UNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES

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1 UNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES IZAK BROERE AND EWA DRGAS-BURCHARDT Abstract. A graph property s any class of graphs that s closed under somorphsms. A graph property P s heredtary f t s closed under takng subgraphs; t s compostve f for any graphs G 1, G 2 P there exsts a graph G P contanng both G 1 and G 2 as subgraphs. Let H be any gven graph on vertces v 1,..., v n, n 2. A graph property P s H- factorzable over the class of graph propertes P f there exst P 1,..., P n P such that P conssts of all graphs whose vertex sets can be parttoned nto n parts, possbly empty, satsfyng: (1) for each the graph nduced by the th non-empty partton part s n P, and (2) for each and j wth j there are no edges between the th and j th parts f v and v j are non-adjacent vertces n H. If a graph property P s H-factorzable over P and we know the graph propertes P 1,..., P n, then we wrte P = H[P 1,..., P n ]. In such a case, the presentaton H[P 1,..., P n ] s called a factorzaton of P over P. Ths concept generalzes graph homomorphsms and (P 1,..., P n )-colorngs. In ths paper we nvestgate all H-factorzatons of a graph property P over the class of all heredtary compostve graph propertes for fnte graphs H. It s shown that n many cases there s exactly one such factorzaton. Addresses: Izak Broere (1) Department of Mathematcs Unversty of Johannesburg P.O. Box 524 Auckland Park 2006 South Afrca Ewa Drgas-Burchardt (2) Faculty of Mathematcs, Computer Scence and Econometrcs Unversty of Zelona Góra Prof. Z. Szafrana 4a, Zelona Góra Poland Emal addresses: (1) zak.broere@up.ac.za. Present address: Department of Mathematcs and Appled Mathematcs, Unversty of Pretora, Pretora, 0002 South Afrca (2) E.Drgas-Burchardt@wme.uz.zgora.pl Keywords: graph property, heredtary, compostve property, unque factorzaton, mnmal forbdden graphs, reducblty Subject Classfcaton: 05C75, 05C15, 05C35 Abbrevated ttle: Unque factorzaton of graph propertes 1

2 2 IZAK BROERE AND EWA DRGAS-BURCHARDT 1. Introducton Factorzatons of graphs and propertes of graphs were frst consdered through nvestgatons of colorngs of graphs. Many researchers, n dfferent branches of mathematcs, have studed extensons of ths topc such as H-colorng (to be homomorphc to a graph H), (P 1,..., P n )-colorng and jons n lattces of graph propertes [1, 2, 3, 4]. All these concepts are related to parttonng problems. The class of n-colorable graphs can be vewed as the class of those graphs G for whch there exsts a partton of ts vertex set nto sets V 1,..., V n, so that each nduced subgraph G[V ] s an edgeless graph. Allowng other possbltes than edgeless graphs, say havng a graph G[V ] wth property P, we obtan the defnton of the class of (P 1,..., P n )-colorable graphs. On the other hand, restrctng requrements n the defnton of n-colorablty by requrng that edges between parts V and V j are forbdden n G for some lsted pars of ndces from the set {1,..., n}, we obtan the defnton of H-colorablty (to be homomorphc to the graph H). In ths case H s a graph wth vertex set {1,..., n} satsfyng that and j are non-adjacent n H f and only f edges between V and V j are forbdden. In ths paper we put together these generalzatons of n-colorngs and we nvestgate the classes of graphs whch have the specal parttons whch are descrbed below. A graph property P s any non-empty somorphc closed subclass of the class of all fnte non-somorphc graphs havng at least one vertex I. The set I s called a trval graph property. A graph property P s heredtary f t s closed under takng subgraphs, P s addtve f for any two graphs G 1, G 2 P the (dsjont) unon of G 1 and G 2 s also n P and P s compostve f for any two graphs G 1, G 2 P there exsts a graph n P contanng both G 1 and G 2 as subgraphs. Note that every addtve graph property s compostve. We denote the class of all heredtary graph propertes and all heredtary and compostve graph propertes by L and L c respectvely. For nstance, the graph propertes S k, O k, where S k = {G I : (G) k} and O k = {G I : G s k-colorable} are heredtary and addtve (and thus compostve) for each k N. On the other hand, a graph property that contans K 3, all forests and all graphs obtaned by takng the dsjont unon of K 3 and a forest s heredtary and compostve but not addtve. Moreover, Oc k = {G I : G s a connected k-colorable graph} s an example of a graph property whch s compostve but not addtve and not heredtary. Let n 2 and let H be a graph wth vertex set {v 1,..., v n }. Let P 1,..., P n be graph propertes, that s, classes of graphs closed under somorphsms. We defne a graph property P to be H-factorzable over a class of graph propertes P, f there exst propertes P 1,..., P n P such that P conssts of all the graphs whose vertex sets can be parttoned nto n parts, possbly empty, such that: (1) for each the graph nduced by the th non-empty partton part s n P, and (2) for each and j wth j there are no edges between the th and j th parts f v, v j are non-adjacent vertces n H. If P s H-factorzable wth factors P 1,..., P n, then we wrte P = H[P 1,..., P n ]. In vew of ths defnton t s easy to see that the classes of n-colorable, (P 1,..., P n )-colorable and H-colorable graphs are K n [O,..., O], K n [P 1,..., P n ] and H[O,..., O] respectvely where O s the graph property to be edgeless and K n s the complete graph on n vertces. Moreover, K n [P 1,..., P n ], where K n denotes the complement of K n, s the jon of propertes P 1,..., P n n the lattce of heredtary graph propertes closed under takng dsjont unons [4].

3 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 3 H-factorzablty was frst consdered for the class of addtve heredtary graph propertes [5, 6]. For gven H-factorzable graph property P, the unqueness of ts H-factorzaton, and the cardnalty of mnmal forbdden graph famly were nvestgated n these papers. Such problems were earler ntensvely studed for n-colorablty, (P 1,..., P n )-colorablty and H-colorablty [7, 8, 9, 10, 11, 12, 13]. In ths paper we explore the H-factorzablty of the class of compostve heredtary graph propertes. We prove the unqueness of H-factorzatons n ths class based on a unque representaton of a graph n terms of the famly of prme graphs. Ths representaton can be constructed n accordance wth Galla s result [14] whch we dscuss n Secton 2. Secton 2 also ncludes some defntons and observatons. In Secton 3 we ntroduce the basc concept of factorzablty. Besdes specal features of the prme graph K 2, some useful remarks are also formulated. Secton 4 nvestgates the unqueness of an H-factorzaton for a prme graph H. Secton 5 contans the man results of the paper. It s readable mmedately after Secton 3 and famlarzaton wth the defnton of a domnatng vertex whch s gven before Lemma Prelmnares All graphs consdered n ths paper are fnte, undrected and smple. Let G denote a graph wth the vertex set V (G) and the edge set E(G). For a gven v V (G) let N G (v) and deg G (v) denote the open neghborhood of v and the degree of v n the graph G respectvely. The maxmum degree and the mnmum degree n G taken over all vertces of G wll be denoted by (G) and δ(g). We wrte G, K n and C n for the complement of G, a complete graph and a cycle of order n respectvely. For gven graphs G 1,..., G n and a graph H wth V (H) = {v 1,..., v n }, we wll use the symbol H[G 1,..., G n ] to denote the graph whose vertex set s the unon of V (G 1 ),..., V (G n ) and whose edge set conssts of the unon of E(G 1 ),..., E(G n ) wth the addtonal edge set {{x, y} : x V (G ), y V (G j ), {v, v j } E(H)}. To emphasze the specal role of H n ths defnton, we call t the base graph of H[G 1,..., G n ]. Note that K n [G 1,..., G n ] s the dsjont unon of the graphs G 1,..., G n, frequently denoted by G 1 G n. Moreover, we adopt the notaton K n [G,..., G] = ng A graph G s homomorphc to a graph H f there exsts a mappng ϕ : V (G) V (H) such that {u, v} E(G) mples {ϕ(u), ϕ(v)} E(H). Such a mappng s sad to be a homomorphsm from G to H. A bjecton ϕ from V (G) onto V (H) n whch {u, v} E(G) f and only f {ϕ(u), ϕ(v)} E(H) s called an somorphsm from G to H and n such a case we say that G s somorphc to H and we wrte G = H. An somorphsm from G to G s called an automorphsm (of G). We shall use the notaton G 1 G 2 to denote the fact that G 1 s somorphc to a subgraph of G 2 ; n such a case we often say that G 1 s a subgraph of G 2. Let [n] = {1,..., n}. If V (H) = {v 1,..., v n } and V (G) = {x 1,..., x m } and ϕ s a homomorphsm (or somorphsm wth n = m) from H to G, then we often smplfy the notaton by wrtng ϕ : [n] [m] nstead of ϕ : V (H) V (G) and ϕ() = j nstead of ϕ(v ) = x j. Followng [14], a set W of vertces of a graph H s a module n H f for any two vertces x, y W, the equalty N H (x) \ W = N H (y) \ W s satsfed. The trval modules n G are V (G), and the sngletons. A graph havng only trval modules s called prme. We denote by PRIME the subclass of I of prme graphs on at least two vertces. It s

4 4 IZAK BROERE AND EWA DRGAS-BURCHARDT worth mentonng that PRIME contans exactly one complete graph K 2 and exactly one dsconnected graph K 2. Moreover, there s no prme graph on three vertces. We now present the man tool, a result of Galla whch descrbes the modular decomposton of an arbtrary graph, whch s to be used n ths paper. Theorem 1. [14] Let G be any graph wth at least two vertces. Then exactly one of the followng condtons holds: (1) G s dsconnected and can be unquely decomposed nto ts connected components, (2) G s dsconnected and can be unquely decomposed nto complements of connected components of G, (3) G and G are connected and there s some U V (G) and a unque partton Π of V (G) such that (a) U 4, (b) the subgraph of G nduced by U s a maxmal prme nduced subgraph of G, and (c) every part S of the partton Π s a module n G wth S U = 1. There does not seem to be a conventon on how to descrbe the modular decomposton of a graph gven n Theorem 1. Thus we gve the most convenent way for the presentaton of our results. Remark 1. Each graph G I on at least two vertces can be unquely constructed by takng V (G) copes of K 1, usng the operaton H[G 1,..., G n ], n 2 wth H a prme base graph and wth G 1,..., G n graphs obtaned n prevous steps. The unqueness s understood to be up to automorphsms of the base graph and features of the specal base graphs K 2 and K 2. We now descrbe how to apply Theorem 1 to buld the prme graph H mentoned n Remark 1 for a gven graph G. If G (G) s dsconnected wth components G 1,..., G s, s 2, then we use the prme graph H = K 2 (H = K 2 respectvely) s 1 tmes. To be more precse, G = K 2 [G 1, K 2 [G 2, K 2 [..., K 2 [G s 1, G s ]...]]] (G = K 2 [G 1, K 2 [G 2, K 2 [..., K 2 [G s 1, G s ]...]]] respectvely) where 1, 2,..., s s an arbtrary permutaton of the numbers from [s]. The structure of G does not depend on the permutaton used but on propertes of K 2 (K 2 ) used as base graphs n an applcaton of the defnton. If none of the above cases hold (.e. f G and G are connected), then we fnd maxmal modules n G dfferent from V (G), say V 1, V 2,..., V p. It s known that V 1, V 2,..., V p are parwse dsjont sets and they form a partton of V (G). The last base graph H used n the constructon has {V 1,..., V p } as vertex set wth two vertces V r and V s adjacent n H f and only f at least one vertex from V r s adjacent n G to at least one vertex from V s, r, s [p]. It s evdent that such H s prme wth at least four vertces and V (H) can play the role of the set U n condton (3) of Theorem 1. At the same tme V 1,... V p s the unque partton Π n Theorem 1. Repeatng the procedure for graphs nduced n G by V 1,..., V p or for components of G (complements of components of G) n the above cases, we obtan the next prme base graphs of the constructon. The procedure termnates f all graphs G[V ], [p] or all components of G (complements of components of G) are somorphc to K 1. Readers famlar wth the noton of the modular decomposton tree of a graph [14, 15] wll see that the descrbed constructon corresponds to ths tree. We focus on the structure between modules and descrbe t from the top n the language of a known product of graphs. Remark 1 mples the followng observaton.

5 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 5 Remark 2. Let G, G 1,..., G n, G 1, G 2,... G m I and H 1, H 2 be prme graphs. If G = H 1 [G 1,..., G n ] = H 2 [G 1,..., G m] then H 1 s somorphc to H 2. Moreover, f H 1 has at least three vertces, then there exsts an automorphsm ϕ of H 1 such that for each [n], the graph G s somorphc to the graph G ϕ(). G s a generatng set for a heredtary graph property P f G P and every graph n P s a subgraph of some G G. A generatng set s ordered f ts elements can be lsted as G 1, G 2,... such that G s a subgraph of G +1 for each permssble. For example, for each G I, the set {G} s a generatng set for the heredtary and compostve graph property consstng of all subgraphs of G. A specal case of a generatng set for P L, the set of P-maxmal graphs denoted M(P), s {G I : G P and G + e / P for each e E(G)}. The followng theorem jonng the notons ntroduced above was verfed n [10]. Theorem 2. [10] For any graph property P, the followng are equvalent: (1) P L c, (2) P has a generatng set; moreover, for any graph L P and any generatng set G for P, the set {G G : L G}, denoted by G[L], s also a generatng set, (3) P has a (fnte or nfnte) ordered generatng set, (4) P has a (fnte or nfnte) ordered generatng set {H 1, H 2,...} n whch H s P-maxmal and t s an nduced subgraph of H +1 for all permssble. Let P 1,..., P n L c, G I and let H be a graph wth V (H) = {v 1, v 2,..., v n }. An H[P 1,..., P n ]-partton of G s defned as a partton (V 1,..., V n ) of V (G) such that the exstence of {x, x j } E(G) wth x V, x j V j, j mples the exstence of {v, v j } E(H) and G[V ] P or V = for [n]. The symbol H[P 1,..., P n ] denotes the class of all graphs possessng an H[P 1,..., P n ]-partton. In other words, H[P 1,..., P n ] conssts of all graphs H[G 1,..., G n ] such that G P, [n], and all ther subgraphs. Evdently the condton P 1,..., P n L c forces that H[P 1,..., P n ] L c. For P = H[P 1,..., P n ] we say that H[P 1,..., P n ] s an H-factorzaton (or, brefly, a factorzaton) of P over L c and P 1, P 2,..., P n are H-factors (or, brefly, factors) of ths factorzaton. Let P 1,..., P n L c, and let H be a graph wth V (H) = {v 1,..., v n }. Next let P = H[P 1,..., P n ] and suppose there exsts a P-maxmal graph G such that each H[P 1,..., P n ]-partton (V 1,..., V n ) of G satsfes V for all [n]. Then H[P 1,..., P n ] s called a proper H-factorzaton of P over L c and the set of all P-maxmal graphs havng the desred property s denoted by M (H[P 1,..., P n ]). For example, let P = P 3 [Q, Q, Q], where {K 2 } and {K 1 } are generatng sets for Q and Q respectvely. Then P 3 [K 2, K 1, K 2 ] M (P 3 [Q, Q, Q]), whch shows that ths P 3 -factorzaton of P s ndeed proper. We now formulate an obvous remark whch appeared n [12] for the case H = K n. Remark 3. Let P 1,..., P n L c and let H be a fxed graph. Every graph G M (H[P 1,..., P n ]) has the form H[G 1,..., G n ] wth G a P -maxmal graph for all [n]. Note that for each graph G M (H[P 1,..., P n ]) and any supergraph G of G n I, we have n each H[P 1,..., P n ]-partton (V 1,..., V n ) of G that V for all [n]. It follows that f H[P 1,..., P n ] s a proper H-factorzaton of P, then M (H[P 1,..., P n ]) s a generatng set for P.

6 6 IZAK BROERE AND EWA DRGAS-BURCHARDT In the lght of Theorem 1 we can see that, f H s a graph on at least two vertces, then each H-factorzaton of a gven graph property P can be vewed as a composton of G- factorzatons wth G beng prme graphs. To get a feelng for argung wth an extendend structure, we start wth the followng lemma. Lemma 1. Let H 1 and H 2 be prme graphs wth V (H 1 ) = {v 1,..., v n }, V (H 2 ) = {v 1,... v m}, and let H 1 [P 1,..., P n ], H 2 [P 1,..., P m] be proper factorzatons of P over L c. Then H 1 = H 2 and consequently n = m. Moreover, f n 3 and G M (H 1 [P 1,..., P n ]) M (H 2 [P 1,..., P n]), and (V 1,..., V n ) s an arbtrary H 1 [P 1,..., P n ]-partton of G, and n) s an arbtrary H 2 [P 1,..., P n]-partton of G, then there exsts an automor- = V ϕ(), [n]. (V 1,... V phsm ϕ : [n] [n] of H 1 such that V Proof. By ponts (4) and (2) of Theorem 2 we can construct a generatng set for P consstng of only P-maxmal graphs from the set M (H 1 [P 1,..., P n ]) M (H 2 [P 1,..., P n]). It follows from Remark 3 that each such graph G satsfes G = H 1 [G 1,..., G n ] = H 2 [G 1,..., G n] and usng Remark 2, by the assumpton that H 1, H 2 are prme, we know that H 1 s somorphc to H 2. Let n 3 and (V 1,..., V n ), (V 1,..., V n) be the assumed parttons of G M (H 1 [P 1,..., P n ]) M (H 2 [P 1,..., P n]). It follows that G = H 1 [G[V 1 ],..., G[V n ]] = H 1 [G[V 1 ],..., G[V n]]. The asserton now follows from Remark C-reducblty For a gven graph H wth V (H) = {v 1,..., v n }, we say that a graph property P L c s H-reducble over L c f there exst non-trval propertes P 1,..., P n L c such that H[P 1,..., P n ] s a proper H-factorzaton of P. Otherwse, P s called H-rreducble over L c. If H[P 1,..., P n ] = H[P 1,..., P m] = P and there exsts an automorphsm ϕ of H satsfyng P ϕ() = P we do not dstngush between these two H-factorzatons. Let C PRIME. A graph H on at least two vertces s sad to be a C-graph f all base graphs used n the unque constructon of H n accordance wth Remark 1 are from C. Moreover, we adopt the conventon that for each C PRIME, the graph K 1 s a C-graph. For nstance, beng a co-graph s equvalent to beng a {K 2, K 2 }-graph. A graph property P L c s C-reducble over L c f P s H-reducble over L c for some H beng a C-graph on at least two vertces, otherwse P s called C-rreducble over L c. Each H-factorzaton of P over L c, where H s a C-graph, s sad to be a C-factorzaton of P over L c. A C-factorzaton of P over L c s rreducble f all ts factors are C-rreducble over L c. The frst part of the statement of Lemma 1 allows us to deduce the theorem below. Theorem 3. If a graph property P s {H}-reducble over L c for some prme graph H then P s {H 1 }-rreducble over L c for each prme graph H 1 whch s not somorphc to H. Proof. If P s {H}-reducble and {H 1 }-reducble over L c for prme graphs H and H 1, then H = H 1 by Lemma 1. The problems of {H}-reducblty, over L c and over other classes of graph propertes were earler dscussed only for H = K 2 and H = K 2 [7, 8, 9, 10, 11, 12, 13]. The second one (see [8]) only over the class L a of addtve heredtary graph propertes (whch s closed under

7 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 7 dsjont unons of graphs) because n such a class {K 2 }-reducblty s equvalent to beng a jon of ncomparable propertes from the lattce (L a, ) [1]. Propertes whch are {K 2 }- reducble over a gven class of propertes always have an rreducble {K 2 }-factorzaton over those class of graph propertes unlke {H}-reducblty n general. We support ths observaton by showng examples of heredtary compostve propertes whch are {K 2 }- reducble over L c but whch have no rreducble {K 2 }-factorzatons over L c. In order to do so, we need some defntons. Let C PRIME. A graph property P 1 s C-nfnte over L c f there exsts an nfnte sequence of graphs H n C, N, and an nfnte sequence of heredtary compostve graph propertes P, N, such that H [P j1,..., P jn ] wth P jn = P +1 s a factorzaton of P over L c (here n s the number of vertces of H ). Otherwse, P 1 s C-fnte over L c. For example the propertes S 2 = {G I : (G) 2}, P = {G I: each component of G s a subgraph of C 3 } are {K 2 }-nfnte over L c. It s so because S 2 = K 2 [Q 3, K 2 [Q 4, K 2 [Q 5,...]]] and P = K 2 [Q, K 2 [Q, K 2 [Q,...]]], where Q = {G I: each component of G s a subgraph of C } and Q = {G I: G s a subgraph of C 3 }. It s very nterestng that S 2 = K 2 [Q 3, R], where R = {G I: for each component G of G there s N \ {1, 2, 3} such that G C }. Ths means that S 2 has a proper K 2 -factorzaton over L c and hence t s {K 2 }-reducble over L c. On the other hand P s {K 2 }-rreducble over L c. To observe t, suppose that for some s N, s 2, the factorzaton P = K s [P 1,..., P s ] over L c s proper. Now, for each N there s a fnte number j such that j C 3 / P, otherwse P = P, contrary to the fact that the factorzaton s proper. It means that kc 3 / P, for k = [s] j, whch s mpossble by the defnton of P. Remark 4. Let C 1 C 2 PRIME. If a graph property P s C 2 -fnte over L c, then t s C 1 -fnte over L c. Usng ths fact, we can now conclude that both S 2 and P are C-nfnte over L c for all C contanng K 2. In the man part of the paper we study rreducble C-factorzatons of P over L c. To obtan such a factorzaton we use a procedure n whch a gven reducble graph property s decomposed untl we obtan rreducble factors. It s not obvous that such a procedure always termnates after fntely many steps, but the followng fact can mmedately be observed. Remark 5. Let C PRIME. Each C-fnte over L c graph property P has an rreducble C-factorzaton over L c. We now consder the C-nfnteness of a graph property on a set C. Theorem 4. Let H, N, be an nfnte sequence of graphs n PRIME and let P, N, be heredtary graph propertes such that H [P 1 = P +1,..., P n ] s a factorzaton of P over L. If there exsts an nfnte subsequence H n, N, such that H n K 2 for each, then P 1 s the trval graph property I. Proof. We shall show that K q P 1 for each q N whch mples that P 1 = I. Let { k } k N be an nfnte ncreasng sequence of postve ntegers such that H k K 2, k N. Evdently K 1 P for each N, and especally K 1 P q. Moreover, for each j > k we have P j P k, hence K 1 P q Because P q 1 has the H q 1 -factorzaton as assumed

8 8 IZAK BROERE AND EWA DRGAS-BURCHARDT and δ(h q 1 ) 1 (snce H q 1 s dfferent from K 2 ), t follows that K 2 P q 1. By the same reasonng K 3 belongs to P q 2, etc. Hence K q s an element of P 1 and fnally K q P 1. Corollary 1. If K 2 / C PRIME and P L c s non-trval, then P s C-fnte over L c. Usng Remark 5, we can now also deduce the followng result. Corollary 2. If K 2 / C PRIME and P L c s non-trval, then P has an rreducble C-factorzaton over L c. We now turn our attenton to questons about the rreducblty of C-factorzatons for sets C of prme graphs contanng K 2. Before we dscuss ths, we wll prove a lemma whch wll be useful n later sectons too. Lemma 2. Let P L c be {K 2 }-rreducble over L c and let G 1 and G 2 be two connected graphs whch are ncomparable n (I, ). If G 1 G 2 P, then there exsts a connected graph G P such that G 1 G 2 G. Moreover, f P L c s {K 2 }-rreducble and {K 2 }-fnte over L c, then for any two connected graphs G 1 and G 2 satsfyng G 1 G 2 P there s a connected graph G P such that G 1 G 2 G. Proof. In the frst part of the proof we only assume that G 1 and G 2 are arbtrary connected graphs n P. Suppose that P does not contan any connected graph G satsfyng G 1 G 2 G. Consder a (fnte or nfnte) ordered generatng set G = {H 1, H 2, H 3...} for P, such that G 1 G 2 H 1 H 2...; ts exstence s guaranteed by Theorem 2. By assumpton each graph H s dsconnected and we can assume that t has a form H = H 1 H 2 H s, s 2, J, wth H j a connected component of H for all permssble parameters, j. Then we defne a (fnte or nfnte) graph T wth V (T ) = {H j : J, j [s ]} and E(T ) constructed n the followng way: for each J for whch + 1 J, choose an arbtrary embeddng of H n H +1 (as a subgraph). Wth respect to that embeddng for ths fxed J and any j [s ], there s a k [s +1 ] such that H j Hk +1. For each such set of ndces we have an edge {H j, Hk +1} n E(T ). Note that T s a forest. It follows because H j has to be contaned n exactly one component of H +1 for fxed, j. A crucal observaton s that T has at least two components. Ths s a consequence of the fact that G 1 and G 2 are contaned n dfferent components of H for each J. Moreover, at least two components of T contan vertces whch are components of H 1. Assume that T has components {T q } q U. Clearly, U can be fnte or nfnte. Let W q = {j [s ] : H j s a vertex of the component T q of T }. We now consder two cases: (1) G 1, G 2 are ncomparable n (I, ) and P s {K 2 }-rreducble over L c. Let U 1 be the set of ndces q n U such that at least one vertex of V (T q ) contans G 1 as a subgraph. Recall that our constructon of T mples that G 2 s not a subgraph of any vertex n V (T q ), where q U 1. Next, let for all, W 1 = q U1 W q and W 2 = q U\U1 W q. Now we construct two propertes: P 1 = {G I : G j W 1 H j, J}, and P 2 = {G I : G j W 2 H j, J}. The defntons mply that P 1, P 2 are heredtary. Further observe that for any G 1, G 2 P 1 one can fnd ndces 1, 2 J, say > 2, such that G s j W 1 s H j s, s {1, 2}. Thus j W 1 1 H j 1 contans both G 1, G 2 as subgraphs and hence P 1 s

9 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 9 compostve. Smlarly we can prove an analogous fact for P 2. By the defnton, P = K 2 [P 1, P 2 ] and ths {K 2 }-factorzaton of P over L c s proper because G 1 G 2 / P 1 P 2. Ths contradcts the {K 2 }-rreducblty of P. (2) To prove the last part of the asserton t s enough to consder the case n whch G 1 G 2 and P s {K 2 }-rreducble and {K 2 }-fnte over L c. For each permssble q, consder the graph property P q = {G I : G j W q Hj, J}. Ths defnes countable many (fnte or nfnte) propertes, each of whch s heredtary and compostve. In the nfnte case P = K 2 [P 1, K 2 [P 2, K 2 [P 3, K 2 [...]]]], whch means that P s {K 2 }-nfnte, a contradcton. In the fnte case we consder the numbers k, p defned n the followng way k = max{m N : mg 2 P}, and p = U. Evdently 1 k p. Let T q1,..., T qk be the components of T satsfyng that G 2 s contaned n at least one of the vertces of V (T q ), [k]. Frst we suppose that k 2. Hence, assumng that W = m [k]\{1} W qm we have P = K 2 [Q 1, Q 2 ], where Q 1 = {G I : G j W H j, J}, Q 2 = {G I : G j [s ]\W Hj, J}. The last factorzaton s proper because kg 2 P and kg 2 / Q 1 Q 2. Usng the same arguments as before, we can see that the propertes Q 1, Q 2 are heredtary and compostve, whch means that the factorzaton s over L c. For k = 1 we construct the proper factorzaton P = K 2 [Q 1, Q 2 ], where Q 1 = {G I : G q j W 1 H j, J}, Q 2 = {G I : G j [s ]\W q 1 H j, J}. Ths factorzaton, as before, has the desred propertes. Usng Lemma 2 we can check at once that S k, for k 2, s {K 2 }-reducble over L c because there s no connected graph n S k contanng as subgraphs two graphs whch are ncomparable n (I, ), connected and k-regular. Assume that S k, k 2, has rreducble proper {K 2 }-factorzaton K s [Q 1,..., Q s ], s 2, over L c. Usng Lemma 2 once agan wth the {K 2 }-rreducblty of Q over L c for each permssble, we can observe that for all s+1 graphs G 1,..., G s+1 whch are ncomparable n (I, ), k-regular and connected, we have G 1 G s+1 / K s [Q 1,..., Q s ] = S k, whch s n contrast to the defnton of S k. Now we consder the graph property R = K 2 [S k, S k ]. Obvously, the gven factorzaton of R over L c s proper because for every two k-regular graphs G 1 and G 2, the graph K 2 [G 1, G 2 ] R and K 2 [G 1, G 2 ] / S k. Clearly, by defnton, R s {K 2 }-reducble over L c. Hence, from Theorem 3 t s {K 2 }-rreducble over L c. Further, by the same reasonng as before R s {K 2, P 5, K 2 }-reducble over L c and stll does not possess an rreducble proper {K 2, P 5, K 2 }-factorzaton over L c. Moreover, t s {K 2, P 5, K 2 }-nfnte and {K 2 }-fnte over L c. In fact, R s C-nfnte for each C contanng {K 2, K 2 }. In general, note that f P s C-fnte over L c and P = H[P 1,..., P n ] s ts C-factorzaton over L c, then each P s C-fnte over L c. On the other hand, C-nfnteness of each P does not mply C-nfnteness of P. 4. Unqueness Lemma 3. If P L c s {K 2 }-rreducble and {K 2 }-fnte over L c, then the set of all connected graphs n P s compostve.

10 10 IZAK BROERE AND EWA DRGAS-BURCHARDT Proof. Suppose that G 1, G 2 are connected graphs n P havng no connected common supergraph n P. Hence, because P L c, we have G 1 G 2 P. Applyng Lemma 2 we obtan a contradcton. The asserton of the next lemma was stated earler n [8] for propertes, whch are {K 2 }-rreducble over L a ; the proof presented here mtates ther proof. Lemma 4. Let P L c be {K 2 }-rreducble and {K 2 }-fnte over L c. Then there exsts a (fnte or nfnte) ordered generatng set of connected graphs for P. Proof. Let {G : J}, (J = [s] for some natural s or J = N) be the set of all connected graphs n P. Let H 1 = G 1 and, for each J \ {1}, let H be a connected graph whch contans H 1 and G as subgraphs smultaneously. The exstence of such graphs s guaranteed by Lemma 3. Of course H 1 H Moreover, {H : J} s a generatng set for P because, n accordance wth Lemma 2, each dsconnected graph n P has a connected supergraph and, by the constructon, each connected graph n P s contaned n H for some J. Lemma 5. Let P be a non-trval graph property whch s {K 2 }-fnte over L c and, for n 2, let K n [P 1,..., P n ] be an rreducble proper {K 2 }-factorzaton of P over L c. Then there s no proper K m -factorzaton of P over L c wth m > n. Proof. Suppose, to the contrary, that such a factorzaton K m [P 1,..., P m] exsts. As we noted earler propertes P and P j are then {K 2 }-fnte over L c for all permssble, j. Let j [n] be fxed and let {Hs j : s L j }, (L j = N or L j = [k j ] for some natural number k j ) be an ordered generatng set for P j that conssts of only connected graphs. The exstence of such sets follows from Lemma 4. Let W = N f for at least one j, L j = N and let W = [max{k j : j [n]}], otherwse. Put Hs j = H j L j for s W, s > L j, j [n]. It s obvous that {H = H 1 H n : W } s a generatng set for P. Now, f a graph G has a form G 1 G 2 G n wth G P, [n], then there exst ndces m, [n], satsfyng G Hm and m = max{m : [n]} and therefore G H m. Snce K m [P 1,..., P m], m > n, s a proper factorzaton of P, there s a P-maxmal graph G satsfyng the defnton. One can then fnd an ndex q such that G H q whch s a supergraph of G and shows that the correspondng factorzaton of P s proper too. Ths means that H q has at least m components whch gves n m, a contradcton. Now we are n a poston to prove the frst unque factorzaton type theorem. Theorem 5. Each non-trval graph property P L c whch s {K 2 }-fnte over L c has a unque proper rreducble {K 2 }-factorzaton over L c. Proof. By Lemma 5 t s enough to consder only the proper factorzatons of P wth the same number of factors. Let K n [P 1,..., P n ], K n [P 1,..., P n], be two rreducble proper {K 2 }-factorzatons of P over L c. We shall show that there exsts a permutaton σ : [n] [n], such that P = P σ(). Let {H j = Hj 1 Hj n : j W } be a generatng set for P, whch was constructed as n Lemma 5 usng the ordered generatng sets {Hj, j W } for P, [n], of connected graphs. Wthout loss of generalty we can assume by Theorem 2 that each graph H j s a supergraph of graphs L 1 and L 2 smultaneously, where L 1 and L 2 are graphs showng that the factorzatons K n [P 1,..., P m] and K n [P 1,..., P n ] are proper factorzatons of P respectvely. It mples that n an arbtrary K n [P 1,... P n]-partton

11 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 11 of H j all partton parts are non-empty. It means that for each j W there exsts a permutaton σ j : [n] [n] satsfyng Hj P σ j (). The set of all permutatons of the set [n] s fnte. If W s nfnte then at least one permutaton n the sequence {σ j, j W } s repeated nfntely many tmes. In that case let σ be such a repeated permutaton and let σ = σ W f W s fnte. Moreover, let W = {j W : σ = σ j }. Obvously, the set of all H j, satsfyng j W s a generatng set for P. It s so because for each G P there exsts an ndex k such that G H k, and then there exsts an ndex j > k satsfyng σ j = σ. By the same reasonng and the constructon we know that for an arbtrary [n], the set {Hj, j W } creates the generatng set for P. We shall show that P s = P σ(s) for each s [n]. Suppose G P s for fxed s [n]. Then there exsts k W such that G Hk s. But Hs k P σ(s) so that P s P σ(s). Assume to the contrary that X P σ(s) \ P s. Moreover let X be a connected graph n P s whch s not n the graph property K n 1 [P 1,..., P s 1, P s+1,..., P n ] (ts exstence follows from the fact that the factorzaton K n [P 1,..., P n ] s proper, {K 2 }-rreducblty of P s and Lemma 3). Obvously X P σ(s). By the compostvty of P σ(s) there exsts a graph G P σ(s) whch contans both X and X as subgraphs. Thus by Lemma 4 such a graph s contaned n a connected graph Y whch s n the graph property P σ(s). Of course Y P, consder a K n [P 1,..., P n ]-partton of Y. It s easy to see that Y has to mss of the propertes P 1,..., P n because of ts connectvty. By the constructon, and n partcular snce X Y, t must be P s, whch contradcts the assumpton that X / P s. Gven a graph H and two vertces v 1, v 2 of H, we say that v 1 domnates v 2 n H f N H (v 2 ) s a proper subset of N H (v 1 ), v 1 s a domnatng vertex n H f there exsts a vertex v 2 such that v 1 domnates v 2 n H. The next lemma plays a key part n the argument for unqueness n the general case. Lemma 6. Let H be a prme graph wthout any domnatng vertces wth V (H) = {v 1,..., v n }, n 4. Let G be a fxed graph, P 1,..., P n L c, and let (V 1,..., V n ) be an H[P 1,..., P n ]-partton of G M (H[P 1,..., P n ]). Let X k = G [V k ] f G / P k, and let X k be a graph n the graph property P k and contanng both G [V k ] and G as subgraphs f G P k, and let Ĝ = H[X 1,..., X n ]. Then each H[P 1,... P n ]-partton of Ĝ s of the form (V (X ϕ(1) ),..., V (X ϕ(n) )) wth ϕ an automorphsm of H satsfyng G P ϕ() f and only f G P. Proof. Let (W 1,..., W n ) be an arbtrary H[P 1,..., P n ]-partton of Ĝ. Let V k, for k [n], be an arbtrary set contaned n V (X k ) such that G [V k ] X k [V k ] and V k = V k (actually G [V k ] = X k [V k ] because G [V k ] s a P k -maxmal graph). Accordng to Lemma 1 there exsts an automorphsm ϕ of H satsfyng V k W ϕ(k), k [n]. We shall show that V (X k ) = W ϕ(k), k [n]. If not, let v V (X k ) \ V k be a vertex such that v W ϕ(l) and l k. (1) There exsts [n] \ {k, l} such that v s adjacent to v k and t s non-adjacent to v l n H. Snce ϕ s an automorphsm of H, we have that {v ϕ(), v ϕ(l) } / E(H). But, by the constructon of Ĝ, v s adjacent to all vertces of V, where V W ϕ(), a contradcton. (2) N H (v k ) \ {v l } N H (v l ). The facts that H s prme and does not have domnatng vertces mply that {v l, v k } E(H) and the degree of the vertex v k s smaller

12 12 IZAK BROERE AND EWA DRGAS-BURCHARDT than the degree of the vertex v l. Hence, G [V l ] + K 1 Ĝ[V l ] + K 1. Furthermore, Ĝ[V l ]+K 1 s somorphc to Ĝ[V l {v}] and ths graph s an element of P l gvng that (V 1,..., V l {x},..., V k \{x},..., V n ), for an arbtrary x V k, s an H[P 1,..., P n ]- partton of G, contrary to Lemma 1. The number of ndces [n] such that W nduces n Ĝ the graph contanng G s the same as the number of [n] for whch X has the desred property and t equals the number of ndces [n] satsfyng G P. Ths mples the last clam of the theorem. Theorem 6. Let H be a prme graph on at least three vertces none of whch s domnatng n H. If a graph property P has a proper {H}-factorzaton over L c, then t s unque. Proof. Let H[P 1,..., P n ], H[P 1,..., P n] be two dfferent proper H-factorzatons of P over L c. By Theorem 2, there s a (fnte or nfnte) ordered generatng set G = {G 1, G 2...} n whch each G s P-maxmal and t s an nduced subgraph of G +1. Also, by Theorem 2, t can be assumed that all graphs G are from the set M (H[P 1,..., P n ]) M (H[P 1,..., P n]) wth graphs n (G[L 1 ])[L 2 ] for L 1, L 2 showng that the gven factorzatons are proper. Let (V,1,..., V,n ), ((V,1,... V,n)) be an arbtrary H[P 1,... P n ]-partton of G (H[P 1,... P n]-partton G respectvely). By ϕ 1,..., ϕ s we denote all automorphsms of H. By Lemma 1 there exsts an ndex l = l(g ) [s] satsfyng V,j = V,ϕl (j), j [n]. We shall now show that there exsts an ndex p [s] such that for each permssble we can fnd an H[P 1,..., P n ]-partton (W,1,..., W,n ) of G and an H[P 1,..., P n]-partton (W,1,..., W,n) of G satsfyng the equalty W,j = W,ϕp(j), j [n]. If G s the fnte r-element set then we put p = r and W,j = V (G ) V r,j, j [n] f G s somorphc to H[V (G ) V r,1,..., V (G ) V r,n ] and W,j = V (G ) V r,j, j [n] f G s somorphc to H[V (G ) V r,1,..., V (G ) V r,n]. If G s nfnte then p equals an arbtrary nfntely many tmes repeated number n the sequence l(g 1 ), l(g 2 ),... whch s well descrbed because l(g ) [s] for all ndces. In ths case W,j = V (G ) V t,j, j [n], where t s an arbtrary ndex greater then such that l(g t ) = p and G s somorphc to H[V (G ) V t,1,..., V (G ) V t,n ]. Smlarly W,j = V (G ) V t,j, j [n], where t s the same ndex as n the prevous lne and G s somorphc to H[V (G ) V t,1,..., V (G ) V t,n]. In both cases dependng on the fnteness of G the equalty W,j = W,ϕp(j) holds, for j [n] and all permssble ndces. We contnue to show that P j = P ϕp(j) for all j [n]. Let for the fxed j [n], Q j be a graph property for whch {G [W,j] : all permssble } s a generatng set. Of course Q j P ϕp(j) and Q j P j. Let us take a look at the nverse nclusons. Frst suppose that there s an L P j \ Q j for selected j [n]. Let q be fxed and (W q,1,..., W q,n) be a partton as n the paragraph above. Smlarly to Lemma 6, let Y k = G q [W q,k ] f L / P k and Y k be a graph n the graph property P k whch contans both G q[w q,k ] and L as subgraphs f L P k. Put Ĝ = H[Y 1,..., Y n ]. It then follows by Lemma 6 that an arbtrary H[P 1,... P n ]-partton of Ĝ s of the form (V (Y ϕ(1)),..., V (Y ϕ(n) )) wth ϕ beng an automorphsm of H satsfyng L P ϕ(j) f and only f L P j. The next observaton s that, because G s a generatng set for P and Ĝ P, there exsts an ndex r such that Ĝ G r. For each H[P 1,..., P n]-partton of G r we can construct the H[P 1,..., P n]- partton of Ĝ takng as a partton part the ntersecton of the prevous one wth V (Ĝ). Thus, by Lemma 6, n each H[P 1,..., P n ]-partton of G r, especally n (W r,1,..., W r,n),

13 UNIQUE FACTORIZATION OF GRAPH PROPERTIES 13 we obtan L G r [Y r,j ], whch proves that L Q j, contrary to the assumpton. Thus there s no such L and P j Q j. The case where P ϕp(j) Q j s dealt wth smlarly. These arguments yeld P j = P ϕp(j) for all j [n]. We now show wth an example that the assumpton about the non-exstence of domnatng vertces n H s necessary. Consder the prme graph H = P 4, the path on four vertces, whch has domnatng vertces. Let V (P 4 ) = {v 1, v 2, v 3, v 4 } and E(P 4 ) = {{v, v +1 } : [3]} and let P 1 = {G I : each component of G s a subgraph of a cycle C 3 or C 5 }, P 2 = P 4 = {K 1 }, P 3 = {G I : each component of G s a subgraph of a cycle C 5 or C 4 }. Consder a factorzaton P 4 [P 1, P 2, P 3, P 4 ] of P. It s easy to see that the graph P 4 [C 3, K 2, C 4, K 1 ] confrms that the factorzaton s proper. However, P does not have a unque proper factorzaton snce another representaton could be P = P 4 [P1, P 2, P 3, P 4 ], where P1 = {G I : each component of G s a subgraph of C 3 }. The next theorem complements Theorems 5 and 6. Theorem 7. [9, 10] Each graph property P L c has a unque rreducble proper {K 2 }- factorzaton over L c. 5. Man results We now gve the two man theorems of the paper and gve one short proof for both of them obtaned by applyng our earler results. Theorem 8. Let C be a famly of prme graphs wthout domnatng vertces and let K 2 / C. Each non-trval graph property P L c has a unque proper rreducble C-factorzaton over L c. Theorem 9. Let C be a famly of prme graphs wthout domnatng vertces. If a nontrval graph property P L c s C-fnte over L c, t has a unque proper rreducble C- factorzaton over L c. Proof. By Corollares 1, 2 and Remark 5 there exsts at least one rreducble proper C- factorzaton of P over L c. Assume that H 1 [P1, 1..., P n 1 1 ], H 2 [P2, 1..., P n 2 2 ] are two dfferent rreducble proper C- factorzatons of P over L c. If at least one of the graphs H 1 and H 2 s somorphc to K 1, then P s C-rreducble over L c and P = P 1 = P 1. Next, suppose that V (H 1 ) V (H 2 2. Remark 1 mples that H s ether dsconnected wth components H 1, H 2,...,H s (ths case s possble only n the proof of Theorem 9) or H s dsconnected wth components H 1, H2,, Hs or H = G [H 1,..., H s ] wth G C \ {K 2, K 2 }, {1, 2}. Of course H j s a C-graph for j [s ] and {1, 2}. In each case we relable the vertces of H such that V (H ) = {v 1,..., v n } and V (H 1 ) = {v 1,..., v V (H1 ) }, V (H j ) = {v j 1 t=1 V (Ht) +1,..., v j t=1 V (Ht) }, 2 j s. Evdently n = s t=1 V (Ht ). Moreover, the above forces three mutually exclusve possble forms of a proper factorzaton of H [P 1,..., P n ] (of whch only the last two need to be consdered for the proof of Theorem 8): (1) H [P 1,..., P n ] = K s1 [H 1 [P 1,..., P V (H1 ) ],..., H s [P s 1 (2) H [P 1,..., P n ] = K s1 [H 1 [P 1,..., P V (H1 ) ],..., H s [P s 1 (3) H [P 1,..., P n ] = G [H 1 [P 1,..., P V (H1 ) ],..., H s [P s 1 t=1 V (Ht ) +1 t=1 V (Ht ) +1 t=1 V (Ht ) +1,..., P n ]],,..., P n ]],,..., P n ]].

14 14 IZAK BROERE AND EWA DRGAS-BURCHARDT Thus, by Theorem 3, for both the H 1 -factorzaton and the H 2 -factorzaton we have the same possblty (ether (1), or (2) or (3)). Let Q j = Hj [P j 1 t=1 V (Ht ) +1 j t=1,..., P V (Ht ) ] for all permssble, j. In case (1) ((2)), Remark 1, the assumpton about {K 2 }-rreducblty ({K 2 }-rreducblty) of P j (appled for H j = K 1) and the above consderaton lead to the asserton that Q j s {K 2 }-rreducble ({K 2 }-rreducble respectvely) for all permssble, j. Moreover, n case (1), Remark 4 mples that each Q j as a graph property whch s C-fnte over L c, s {K 2 }-fnte over L c. Hence, n accordance wth Theorem 5 (Theorem 7), s 1 = s 2 = s and there exsts a permutaton ϕ satsfyng Q j 2 = Q ϕ(j) 1 for all j [s]. In case (3), by Theorem 3, G 1 s somorphc to G 2 and by Theorem 6 there exsts an automorphsm ϕ of G 1 such that Q j 2 = Q ϕ(j) 1 for all j [s]. We apply the same reasonng for all Q j whch are C-reducble over Lc. Ths procedure termnates by the assumpton that P s C-fnte over L c. Fnally we compose all automorphsms and permutatons obtaned from ths procedure n order to observe that H 1 s somorphc to H 2. At the same tme ths shows that there exsts an automorphsm ϕ of H 1 such that P j 2 = P ϕ(j) 1. References [1] Boroweck, M., Mhók, P.: Heredtary propertes of graphs. n: Kull, V.R. ed., Advance n Graph Theory (Vshawa Internatonal Publcaton), Gulbarga, 41 68, 1991 [2] Cockayne, E.J.: Colour classes for r-graphs. Canad. Math. Bull. 15, (1972) [3] Hell, P. and Nešetřl, J., Graphs and Homomorphsms, Oxford Lecture Seres n Mathematcs and Its Applcatons, Oxford Unversty Press, 2004 [4] Jakubík, J.: On the Lattce of Addtve Heredtary Propertes of Fnte Graphs. Dscussones Mathematcae General Algebra and Applcatons 22, 73 86(2002) [5] Drgas-Burchardt, E.: On unqueness of a general factorzaton of graph propertes. Journal of Graph Theory 62, 48 64(2009) [6] Drgas-Burchardt, E.: Cardnalty of a mnmal forbdden graph famly for reducble addtve heredtary graph propertes. Dscussones Mathematcae Graph Theory 29(2), (2009) [7] Berger, A.J.: Mnmal forbdden subgraphs of reducble graph propertes. Dscussones Mathematcae Graph Theory 21, (2001) [8] Berger, A.J., Broere, I., Moag, S.J.T., Mhók, P.: Meet- and jon-rreducblty of addtve heredtary propertes of graphs. Dscrete Mathematcs 251, 11 18(2002) [9] Farruga, A.: Unqueness and complexty n generalsed colorng, PhD thess, Waterloo, Ontaro, Canada, 2003 [10] Farruga, A., Mhók, P., Rchter, R. Bruce, Semanšn, G.: Factorzatons and characterzatons of nduced-heredtary and compostve propertes. Journal of Graph Theory 49, 11-27(2000) [11] Farruga, A., Rchter, R. Bruce: Unque factorsaton of addtve nduced-heredtary propertes. Dscussones Mathematcae Graph Theory 24, (2004) [12] Mhók, P., Semanšn, G., Vasky, R.: Addtve and heredtary propertes of graphs are unquely factorzable nto rreducble factors. Journal of Graph Theory 33, 44-53(2000) [13] Zverovch, I.E.: r-bounded k-complete bpartte bhypergraphs and generalzed splt graphs. Dscrete Mathematcs 247, (2002) [14] Galla, T.: Transtv orenterbare Graphen. Acta Mathematca Academae Scentarum Hungarcae 18, 25 66(1967) [15] James, L.O., Stanton, R.G., Cowan, D.D.: Graph decomposton for undrected graphs. n: Proceedngs of the Thrd Southeastern Internatonal Conference on Combnatorcs, Graph Theory and Computng (CGTC 72), , 1972