On the fractional chromatic number, the chromatic number, and graph products

Transcription

1 O the fractioal chromatic umber, the chromatic umber, ad graph products Sadi Klavžar 1 Departmet of Mathematics, PEF, Uiversity of Maribor, Koroška cesta 160, 2000 Maribor, Sloveia sadi.klavzar@ui-lj.si Hog-Gwa Yeh 2 Departmet of Applied Mathematics, Natioal Uiversity of Kaohsiug, Kaohsiug, 811, Taiwa. hgyeh@uk.edu.tw Abstract It is show that the differece betwee the chromatic umber χ ad the fractioal chromatic umber χ f ca be arbitrarily large i the class of uiquely colorable, vertex trasitive graphs. For the lexicographic product G H it is show that χ(g H) χ f (G) χ(h). This boud has several cosequeces, i particular it uifies ad exteds several kow lower bouds. Lower bouds of Stahl (for geeral graphs) ad of Bollobás ad Thomaso (for uiquely colorable graphs) are also proved i a simple, elemetary way. Key words: chromatic umber, fractioal chromatic umber, graph product, uiquely colorable graph. 1 Itroductio A graph product is defied o the Cartesia product of the vertex sets of the factors, while its edges are determied by the edge sets of the factors. There 1 Supported by the Miistry of Sciece ad Techology of Sloveia uder the grat Supported i part by the Natioal Sciece Coucil uder grat NSC M Preprit submitted to Elsevier 5 October 2010

2 are 256 such products. The four most importat of them the Cartesia, the direct, the strog, ad the lexicographic oe are called the stadard graph products ad are defied below. For algebraic ad other reasos for the selectio of these four products to be the stadard oes, see the book [10]. For graphs G ad H, let G H, G H, G H ad G H be the Cartesia, the direct, the strog, ad the lexicographic product of G ad H, respectively. The vertex set of ay of these products is V (G) V (H). Vertices (a, x) ad (b, y) are adjacet i G H wheever ab E(G) ad x = y, or a = b ad xy E(H); are adjacet i G H wheever ab E(G) ad xy E(H); are adjacet i G H wheever ab E(G) ad x = y, or a = b ad xy E(H), or ab E(G) ad xy E(H); are adjacet i G H wheever ab E(G), or a = b ad xy E(H). Aother graph product that will be metioed is the disjuctive product G H (called the iclusive product i [4,15] ad the Cartesia sum i [3]) i which vertices (a, x) ad (b, y) are adjacet wheever ab E(G) or xy E(H). For the Cartesia product Sabidussi [18] showed that for ay graphs G ad H, χ(g H) = max{χ(g), χ(h)}. For the direct product it is easy to see that χ(g H) mi{χ(g), χ(h)}. Hedetiemi [8] cojectured i 1966 that for ay graphs G ad H, χ(g H) = mi{χ(g), χ(h)}. Although the cojecture was widely approached, cf. recet survey [22], oly special cases have bee solved affirmatively. Oe of the reasos for the ivestigatio of the chromatic umber of the lexicographic product is its close relatio to the fractioal chromatic umber, χ f, cf. [19], ad to the cocept of the th chromatic umber, cf. [20,21]. I additio, the chromatic umber of lexicographic products form a importat tool i the theory of approximatio algorithms for the chromatic umber of a graph [15], while i [4] the disjuctive product is used. For more results o the chromatic umber of graph products see [12]. I Sectio 2 we show, usig the graphs (K C 2k+1 ) K 3 1, that the differece betwee the chromatic umber ad the fractioal chromatic umber ca be arbitrarily large i the class of uiquely colorable, vertex trasitive graphs. The we prove that for ay graphs G ad H, χ(g H) χ f (G)χ(H) ad cosider several cosequeces of this fact. I the fial sectio we preset two simple, elemetary proofs of theorems of Stahl ad of Bollobás ad Thomaso. The graphs cosidered are fiite ad simple. As usually, χ(g) deotes the chromatic umber of G ad α(g) its idepedece umber. We will mostly cosider the chromatic umber of the lexicographic product. Clearly, for ay graphs G ad H we have χ(g H) χ(g) χ(h). It is more difficult to obtai a good lower boud for χ(g H). I the rest we will extesively use the followig fudametal result due to Geller ad Stahl: Theorem 1 [6] If χ(h) =, the for ay graph G, χ(g H) = χ(g K ). 2

3 A graph G is called uiquely -colorable if ay -colorig of G determies the same partitio of V (G) ito color classes. We will apply the followig result of Greewell ad Lovász: Theorem 2 [7] If G is a coected graph with χ(g) >, the G K is uiquely -colorable. The fractioal chromatic umber χ f (G) of G is defied as follows. Let I(G) be the set of idepedet sets of a graph G. A fractioal colorig of G is a mappig f : I(G) [0, 1] such that for each vertex v of G we have I I,v I f(v) 1. The weight w(f) of the fractioal colorig f is defied as w(f) = I I f(i). The χ f (G) is the miimum of the weights of fractioal colorigs of G. Note that for ay graph G, χ(g) χ f (G). Gao ad Zhu proved: Theorem 3 [5] For ay graphs G ad H, χ f (G H) = χ f (G) χ f (H). Aalogous result for the disjuctive product is give i [4]. Fially, circulat graphs are defied as follows. Let N be a set of ozero elemets of Z k such that N = N. The circulat graph G(k, N) has vertices 0, 1,..., k 1 ad i is adjacet to j if ad oly if i j N, where the arithmetic is doe mod k. Note that circulat graphs are vertex trasitive. 2 Products ad fractioal chromatic umber I this sectio we treat the iterplay betwee the fractioal chromatic umber ad graph products. We first use the direct product ad the lexicographic oe to show that the differece betwee the chromatic umber ad the fractioal chromatic umber ca be arbitrarily large i the class of uiquely colorable, vertex trasitive graphs. The we observe that for ay graphs G ad H we have χ(g H) χ f (G)χ(H). We list some cosequeces of this boud ad demostrate that it exteds previously kow lower bouds. We begi with the followig well-kow lemma, cf. [10]. Lemma 4 For ay graphs G ad H, (i) (ii) [11,17] α(g H) max{α(g) V (H), α(h) V (G) }, [6] α(g H) = α(g)α(h). I a uiquely colorable, vertex trasitive graph G, all color classes of the uique colorig are of the same size. This observatio might give a feelig that the chromatic umber ad the fractioal chromatic umber coicide o such graphs. (This is clearly ot the case for graphs that are oly vertex 3

4 trasitive, for istace, χ(c 2k+1 ) = 3 ad χ f (C 2k+1 ) = 2 + 1/k.) However, we have the followig: Theorem 5 For ay iteger 2 there exists a uiquely colorable, vertex trasitive graph G, such that χ(g) χ f (G) > 2. PROOF. Let 2 ad let k be a arbitrary iteger >. Set G = (K C 2k+1 ) K 3 1. The lexicographic product ad the direct product of vertex trasitive graphs is vertex trasitive, cf. [10], hece so is G. From Theorem 1 we ifer that χ(k C 2k+1 ) = 3. Therefore, G is uiquely colorable by Theorem 2. It is well-kow that Hedetiemi s cojecture holds for complete graphs (see [1,22]), hece χ(g) = 3 1. Moreover, it is kow (ad easy to prove) that for a vertex trasitive graph G we have χ f (G) = V (G) /α(g). Sice α(g) α(k C 2k+1 )(3 1) by Lemma 4 (i), we fid out, usig Lemma 4 (ii), that α(g) k(3 1). Now, χ f (G) (2k + 1)(3 1) k(3 1) = 2 + k < Thus we coclude that χ(g) χ f (G) > 3 1 (2 + 1) = 2. We ow give a oliear lower boud for the chromatic umber of the lexicographic product of graphs. Although its proof is quite short, it exteds some previously kow lower bouds. Theorem 6 For ay graphs G ad H, χ(g H) χ f (G)χ(H). PROOF. Let χ(h) =. The we have: χ(g H) = χ(g K ) (by Theorem 1) χ f (G K ) = χ f (G)χ f (K ) (by Theorem 3) = χ f (G) (as K is vertex trasitive) = χ f (G)χ(H). 4

5 Cosider the circulat graphs G m = G(3m 1, {1, 4,..., 3m 2}). The χ(g m K ) χ f (G m ) = ((3m 1)/m). Therefore χ(g m K ) 3 /m, which is the exact chromatic umber of these graphs, cf. [13]. Sice G H is a spaig subgraph of G H, we ifer that χ(g H) χ(g H). Thus Theorem 6 implies that χ(g H) χ f (G)χ(H). This observatio ca be used to shorte some of the argumets from [3]. Moreover, we have: Corollary 7 For a graph G, the followig coditios are equivalet: (i) χ f (G) = χ(g), (ii) χ(g H) = χ(g)χ(h), for all graphs H, (iii) χ(g H) = χ(g)χ(h), for all graphs H. PROOF. The equivalece betwee (i) ad (ii) is proved i [19]. Sice χ(g)χ(h) χ(g H) χ(g H), we ifer that (iii) implies (ii). Fially (i) implies (iii) by Theorem 6. It is ot difficult to verify (cf. [13]) that χ f (G) = if{χ(g K )/ = 1, 2,...}. Usig this fact, we have aother proof of Theorem 6. Ideed, let χ(h) =. The χ f (G) χ(g K )/ = χ(g H)/χ(H). Let G () = G G G G ad G [] = G G G G, times. Hell ad Roberts showed i [9] that χ f (G) = if χ(g [] ) = if χ(g () ). They first proved χ f (G) = if χ(g [] ) ad the claimed the secod equality by usig the duality theorem of liear programmig. Usig Theorem 6 we give a elemetary proof of the secod equality, that is, without usig the duality theorem of LP. By Theorem 6 we have χ(g (t) )/χ(g (t 1) ) χ f (G), for every t 1. Hece χ(g) 1 χ(g (2) ) χ(g (3) ) χ(g) χ(g (2) ) χ(g () ) χ(g ( 1) ) χ f(g), = 1, 2, 3,... Therefore, if χ(g () ) χ f (G) = if χ(g [] ), ad so we coclude that if χ(g () ) = if χ(g [] ). 5

6 Note that the above argumet is parallel to the proof of Theorem o page 13 of the book [19]. I the ext two corollaries we show that Theorem 6 exteds some kow lower bouds. We first state: Corollary 8 [20] Let G be a obipartite graph. The for ay graph H, χ(h) χ(g H) 2χ(H) +, k where 2k + 1 is the legth of a shortest odd cycle of G. PROOF. We first observe that it suffices to prove the result for G = C 2k+1. As C 2k+1 is vertex trasitive, χ f (C 2k+1 ) = 2 + 1/k. The result ow follows by Theorem 6. Corollary 8 has bee geeralized i [5] to the so-called circular chromatic umber of graphs. Lovász [16] has show that for ay graph G o vertices Therefore we also have: χ f (G) χ(g) 1 + l α(g). Corollary 9 [16] For ay graphs G ad H, χ(g H) χ(g)χ(h) 1 + l α(g). To see that the lower boud of Theorem 6 is i geeral better tha those of Corollaries 8 ad 9 cosider the Hammig graphs H = K K, 2. Let G = H K. Corollary 8 gives χ(g ) 3. Sice α(h ) =, Corollary 9 implies that χ(g ) 2 /(1+l ). Fially Theorem 6 asserts that χ(g ) 2. To coclude the sectio we add that the iequality χ(g H) χ(g)χ f (H) does ot hold i geeral. For istace, χ(c 2k+1 K ) = 2 + /k while χ(c 2k+1 ) χ f (K ) = 3. This example is a ice illustratio of the fact that the lexicographic product is ot commutative. 6

7 3 Two simple ad short proofs I this sectio we preset simple ad elemetary proofs of two further lower bouds. The mai advatage of the preseted proofs is that they are coceptually simpler tha the existig oes. The first result is due to Stahl [20], see also [14]. Theorem 10 [20] If G has at least oe edge, the for ay graph H, χ(g H) χ(g) + 2χ(H) 2. PROOF. By Theorem 1 it suffices to prove the boud for H = K. Let V (K ) = {1, 2,..., }, χ(g K ) = l, ad let c be a l-colorig of G K. Set U = {v V (G) ; c(v, i) 2 for i = 1,..., }. The U ca be partitioed ito idepedet sets U = {v U ; c(v, i) = 1 for some i} ad U \ U. Hece U iduces a bipartite subgraph of G. For a vertex u V (G) \ U let i u be a vertex of K such that c(u, i u ) The the fuctio φ : V (G) \ U {2 + 1,..., l} give by φ(u) = c(u, i u ) is a (l 2)-colorig of G \ U. For if uu E(G) the (u, i u ) ad (u, i u ) are adjacet i G K ad hece c(u, i u ) c(u, i u ). We coclude that χ(g) (l 2) + 2. Bollobás ad Thomaso improved Theorem 10 for the case of uiquely colorable graphs: Theorem 11 [2] If G is uiquely m-colorable graph, m > 2, the for ay graph H with at least oe edge, χ(g H) χ(g) + 2χ(H) 1. PROOF. It suffices to cosider the case χ(g) = m ad H = K, 2. Let V (K ) = {1, 2,..., }. Suppose o the cotrary that χ(g K ) χ(g) + 2χ(H) 2, ad let c be a (m + 2 2)-colorig of G K. Let U, U ad φ be defied as i the proof of Theorem 10, where the vertices of U (resp. U \ U ) receive color 1 (resp. color 2). We claim that c(u, i) for ay u V (G) \ U ad ay i V (K ). Assume o the cotrary that c(u, i) 2. We may without loss of geerality assume that c(u, i) = 1 (for otherwise we ca redefie U accordigly). But the we ca recolor u with color 1 ad still have a m-colorig of G, which is ot possible sice G is uiquely colorable. 7

8 Sice {c(u, k) : u V (G) \ U ad k V (K )} (m + 2 2) 2 = m 2, V (G)\U m 2 ad 2, there exist two distict vertices x, y V (G)\U such that c(x, i) = c(y, j) for some i, j V (K ). The the fuctio c(x, i); u = x, ϕ(u) = c(y, j); u = y, φ(u); otherwise is a m-colorig of G. Now pick r V (K ) \ {i}, the c(x, r); u = x, ϕ (u) = c(y, j); u = y, φ(u); otherwise is aother m-colorig of G differet from ϕ, sice c(x, r) c(y, j) = c(x, i). Ackowledgmet We are grateful to oe of the referees for her/his relevat remarks. I particular, the preset form of Theorem 6 is due to her/him. Refereces [1] S. A. Burr, P. Erdős ad L. Lovász, O graphs of Ramsey type, Ars Combi. 1 (1976) [2] B. Bollobás ad A. Thomaso, Set colourigs of graphs, Discrete Math. 25 (1979) [3] N. Čižek ad S. Klavžar, O the chromatic umber of the lexicographic product ad the Cartesia sum of graphs, Discrete Math. 134 (1994) [4] U. Feige, Radomized graph products, chromatic umbers, ad the Lovász ϑ- fuctio, Combiatorica 17 (1997) [5] G. Gao ad X. Zhu, Star-extremal graphs ad the lexicographic product of graphs, Discrete Math. 152 (1996) [6] D. Geller ad S. Stahl, The chromatic umber ad other parameters of the lexicographic product, J. Combi. Theory Ser. B 19 (1975)

9 [7] D. Greewell ad L. Lovász, Applicatios of product colourig, Acta Math. Acad. Sci. Hugar. 25 (1974) [8] S. Hedetiemi, Homomorphisms of graphs ad automata, Techical Report T, Uiversity of Michiga (1966). [9] P. Hell ad F. S. Roberts, Aalogues of the Shao capacity of a graph, Aals of Discrete Math. 12 (1982) [10] W. Imrich ad S. Klavžar, Product Graphs: Structure ad Recogitio (Wiley, New York, 2000). [11] P. K. Jha ad G. Slutzki, Idepedece umbers of product graphs, Appl. Math. Lett. 7(4) (1994) [12] S. Klavžar, Colorig graph products - A survey, Discrete Math. 155 (1996) [13] S. Klavžar, O the fractioal chromatic umber ad the lexicographic product of graphs, Discrete Math. 185 (1998) [14] S. Klavžar ad U. Milutiović, Strog products of Keser graphs, Discrete Math. 133 (1994) [15] N. Liial ad U. Vazirai, Graph products ad chromatic umbers, i: Proc. 30th A. IEEE Symp. o Foud. of Comp. Sci. (IEEE Comput. Soc. Press, Los Alamos, 1989) [16] L. Lovász L, O the ratio of optimal itegral ad fractioal covers, Discrete Math. 13 (1975) [17] R. J. Nowakowski ad D. Rall, Associative graph products ad their idepedece, domiatio ad colorig umbers, Discuss. Math. Graph Theory 16 (1996) [18] G. Sabidussi, Graphs with give group ad give graph-theoretical properties, Caad. J. Math. 9 (1957) [19] E. R. Scheierma ad D. H. Ullma, Fractioal Graph Theory: A Ratioal Approach to the Theory of Graphs (Wiley, New York, 1997). [20] S. Stahl, -tuple colorigs ad associated graphs, J. Combi. Theory Ser. B 20 (1976) [21] S. Stahl S, The multichromatic umbers of some Keser graphs, Discrete Math. 185 (1998) [22] X. Zhu, A survey o Hedetiemi s cojecture, Taiwaese J. Math. 2 (1998)