Proceedings of the 2014 Federated Conference on Computer Science and Information Systems pp

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1 Proceedings of the 204 Federated Conference on Computer Science Information Systems pp DOI: /204F297 ACSIS, Vol. 2 An efficient algorithm for the density Turán problem of some unicyclic graphs Halina Bielak Institute of Mathematics Maria Curie Skłodowska University Pl. M. Curie-Skłodowskiej Lublin, Pol hbiel@hektor.umcs.lublin.pl Kamil Powroźnik Institute of Mathematics Maria Curie Skłodowska University Pl. M. Curie-Skłodowskiej Lublin, Pol kamil.pawel.powroznik@gmail.com Abstract Let H = (V(H), E(H)) be a simple connected graph of order n with the vertex set V(H) the edge set E(H). We consider a blow-up graph G[H]. We are interested in the following problem. We have to decide whether there exists a blow-up graph G[H], with edge densities satisfying special conditions (homogeneous or inhomogeneous), such that the graph H does not appear in a blow-up graph as a transversal. We study this problem for unicyclic graphs H with the cycle C 3. We show an efficient algorithm to decide whether a given set of edge densities ensures the existence of H in the blow-up graph G[H]. Index Terms blow-up graph; density; Turán density problem; unicyclic graph. I. INTRODUCTION TURÁN [0] stated the first results in extremal graph theory. Then many authors extended this subject formulated similar new Turán density problems. [], [3], [4], [6], [8], [9] [] obtained interesting results for some families of graphs. In this paper we present an algorithm for testing whether a unicyclic graph with a given set of edge densities is a factor (transversal) of a blow-up graph. Our algorithm has the time complexity at most O(n 2 ), where n is the number of vertices of the unicyclic graph. Csikvári Nagy [5] discovered some interesting algorithm for testing whether a tree with a given set of edge densities is a factor of a blow-up graph. We extend their algorithm to the family of unicyclic graphs with the cycle C 3. Now we define some notions notations. Other definitions one can find in [2] [7]. Let H = (V(H),E(H)) be a simple connected graph of order n with the vertex set V(H) the edge set E(H). By P k we denote the path with k vertices. By C k we denote the cycle with k vertices. The set S V(H) is called an independent vertex set if the subgraph of H induced by S has empty set of edges. Let N H (v) = {x V(H) {v,x} E(H)} be the neighbourhood of the vertex v V(H) in the graph H. N H (v) is called the degree of v in V(H). Each vertex of degree in a graph H is called a leaf of the graph H. We say that the graph H is r-regular if each vertex of H has degree r. The set M E(H) is called the matching (or independent edge set) in the graph H if the subgraph of H induced by M is -regular. For a connected graph H we define a blow-up graph G[H] of the graph H as follows. First we replace each vertex i V(H) by an independent set of vertices A i. Throughout this paper A i is called a cluster. Next we connect vertices between the clusters A i A j if i j are adjacent in H, i,j V(H). The graph induced by A i A j in G[H] is a subgraph of a complete bipartite graph. See Fig. 2 Fig. 3 which present examples of a blow-up graphs G[H] of the graph H presented in Fig.. For any two clusters we define the density between them by the following formula d(a i,a j ) = e(a i,a j ) A i A j, where e(a i,a j ) denotes the number of edges between the clusters A i A j. The graph H is a transversal of G[H] if H is a subgraph of G[H] such that we have a homomorphism φ : V(H) V(G[H]) for which φ(i) A i for all i V(H). Other terminology: H is a factor of G[H]. An edge e = {i,j} of the graph H we denote by e = ij. The density Turán problem can be defined as follows. Let us determine the critical edge density, denoted by d crit, which ensures the existence of the subgraph H of G[H] as a transversal. Precisely, assume that all edges e = {i,j} in the graph H satisfy the condition d(a i,aj) > d crit, where i,j V(H). Then, no matter how we construct the blow-up graph G[H], it contains the graph H as a transversal /$25.00 c 204, IEEE 479

2 480 PROCEEDINGS OF THE FEDCSIS. WARSAW, 204 Fig.. The graph H U n,3 with the vertex set V(H) = {x,y,z,v}. On the other words, for any value d < d crit there exists a blow-up graph G[H] such that Fig. 2. An example of the blow-up graph G[H] of the graph H presented in Fig. with a transversal H. d(a i,a j ) > d for all edges ij E(H) it does not contain H as a transversal. This problem was studied in [9]. By [5] we know that it is useful to consider more general problem. Let us assume that for every edge e E(H) a density γ e is given. Now our task is to decide if the set of densities {γ e } e E(H) ensure the existence of the graph H as a transversal or we can construct a blow-up graph G[H] such that d(a i,a j ) γ ij, but it does not induce the graph H as a transversal. This more general setting allows to use inductive proofs (see the proof of Theorem 7). We call this general case as the inhomogeneous condition on the edge densities, while the above condition of having a common lower bound d crit (H) for densities is called the homogeneous case. Let U n,p be a family of unicyclic graphs of order n with the cycle C p. The path P 2 the cycle C 3 are trivial unicyclic graphs for further considerations. In this paper we study the inhomogeneous density Turán problem for unicyclic graphs in the family U n,3, i.e. with the unique cycle C 3 (see Fig. ). Fig. 2 Fig. 3 present two blow-up graphs G [H] G 2 [H] of the graph H presented in Fig.. In both cases we have the following values of the densities between clusters d(a x,a y ) = d(a y,a z ) = 3 20, Fig. 3. An example of the blow-up graph G[H] of the graph H presented in Fig. without a transversal H. where the summation goes over all matchings of the graph H, including the empty matching. Fig. 4 Fig. 5 present the paths P 2, P 4 the unicycle graph H U 6,3 with variables x e assigned to each edge. Fig. 4. Paths P 2 P 4 with variables x e assigned to each edge. d(a x,a v ) = 3 6, d(a y,a v ) = 0. Let us recall the definition of the multivariate matching polynomial of the graph. The polynomial is the useful tool for the proof of our results. Definition. Let H be a graph let x e be the vector of variables x e, e E(H). We define the multivariate matching polynomial F H of the graph H as follows F H (x e,t) = ( x e )( t) M, M M e M Fig. 5. Graph H U 6,3 with variables x e assigned to each edge. By definition of the multivariate matching polynomial we have F P2 (x e,s) = sx, F P4 (x e,s) = s(x +x 2 +x 3 )+s 2 x x 3, F H (x e,s) = s(x +x 2 +x 3 +x 4 +x 5 +x 6 )+ s 2 (x x 4 +x x 5 +x x 6 +x 2 x 6 +x 3 x 6 +x 4 x 6 ) s 3 x x 4 x 6.

3 KAMIL POWROŹNIK, HALINA BIELAK: AN EFFICIENT ALGORITHM FOR THE DENSITY 48 II. SOME RESULTS FOR THE HOMOGENEOUS CASE For the completness of this paper we present some results for the homogeneous Turán density problem in this section. For this case Nagy [9] presented the following lower upper bounds for the critical density d crit. Theorem (Nagy [9]). For a graph H we have ( ) ( ) d crit (H) (H) 2, (H) where (H) is the maximal degree of H. Then Csikvári Nagy [5] improved the upper bound. Theorem 2 (Csikvári Nagy [5]). Let (H) be the largest degree of the graph H. Then we have d crit (H) e(2 (H) ), where e is the base of the natural logarithm. Now let us recall the definition of the matching polynomial of the graph. Definition 2. Let H be a weighted graph with constant weight function w(e) = for all edges e E(H). Then the matching polynomial is defined as n/2 M(H,t) = ( ) k m k (H)t n 2k, k=0 where m k (H) denotes the number of k independent edges in the graph H. Using this polynomial Csikvári Nagy [5] stated the upper bound for the critical density as in Theorem 3. Theorem 3 (Csikvári Nagy [5]). Let (H) be the largest vertex degree in the graph H let t(h) be the largest root of the matching polynomial. Then we have In particular, d crit (H) (t(h)) 2. d crit (H) < 4( (H) ). What is more Nagy [9] showed the exact value of the critical density for trees. Theorem 4 (Nagy [9]). Let T be a tree. Then we have d crit (T) = λ 2 max (T), where λ max (T) is the maximum eigenvalue of the adjacency matrix of the tree. Furthermore, Nagy [9] showed that for the cycle of order n for the path of order n+ the critical densities are equal. Theorem 5 (Nagy [9]). Let C n be a cycle on n vertices P n+ be a path on n+ vertices. Then we have d crit (C n ) = d crit (P n+ ) = 4cos 2 π. n+2 We formulate the followig open problem. Open problem: count the critical density d crit (H) for H U n,p, p 3. III. INHOMOGENEOUS CASE: UNICYCLIC GRAPHS WITH THE CYCLE C 3 In this section we study the inhomogeneous case when graph H U n,3, e.i. H is unicyclic with the cycle C 3 for each edge e E(H) the edge density γ e is given. We extend some results presented in [5], where authors studied the inhomogeneous case for trees proved the following theorem. Theorem 6. (Csikvári, Nagy [5]) Let T be a tree of order n let v be a leaf of T. Assume that for each edge of T a density γ e = r e is given. Let T be a tree obtained from T by deleting the leaf v the edge uv, where u is the unique neighbour of v. Let the edge densities γ e in T be defined as follows γ e = r e, if e is not incident to u, γ e = re r uv, if e is incident to u. Then the set of densities {γ e } e E(T) ensures the existence of the factor T if only if all γ e (0,] the set of densities {γ e } e E(T ) ensures the existence of the factor T. Theorem 6 provides authors of [5] with an efficient algorithm to decide whether a given set of edge densities in tree ensures the existence of a transversal or does not ensure. Their algorithm is presented below as Algorithm T for the completeness of our paper. We extend the algorithm (Algorithm T ) to the family of unicyclic graphs with the cycle C 3. The new algorithm (AlgorithmU n,3 ) is based on the following Theorem 7 proved below by an extension of the method discovered in [5]. Theorem 7. Let H U n,3 be a unicyclic graph of order n with the cycle C 3 assume that for each edge e E(H) a density γ e = r e is given. If the order of H is greater then 3, let v be a leaf of H u be the unique neighbour of v, then let H be a graph obtained from H by deleting the leaf v an edge uv. Let the densities γ e in H be defined as follows γ e = r e, if e is not incident to u, γ e = re r uv, if e is incident to u. If the order of H is equal to 3 (i.e., H is isomorphic to C 3 with V(H) = {a,b,c}), then let H be a graph obtained from H by deleting the vertex a edges ab ac. H is a path P bc. Let the density γ bc in H be defined as follows γ bc = r bc ( r ab )( r ac ).

4 482 PROCEEDINGS OF THE FEDCSIS. WARSAW, 204 Algorithm T Step 0. Let there be given a tree T 0 edge densities γ 0 e. Set T := T 0 r e = γ 0 e. Step. Consider (T,r e ). if V(T) = 2 0 r e < then STOP: the densities γ 0 e ensure the existence of a factor T 0. if V(T) 2 there exists an edge for which r e then STOP: the densities γ 0 e do not ensure the existence of a factor T0. Step 2. if V(T) 3 0 r e < for all edges e E(T) then DO pick a vertex v of degree let u be its unique neighbour. Let T := T v r e, if e is not incident to u, r e = r e r uv, if e is incident to u. Jump to Step with (T,r e ) := (T,r e). Then the set of densities {γ e } e E(H) ensures the existence of the factor H if only if all γ e (0,] the set of densities {γ e} e E(H ) ensures the existence of the factor H. Proof. Let H U n,3 let the set of densities γ e = r e be given for each e E(H). First we prove the following statement: if all γ e are indeed densities they ensure the existence of a factor H, then the original densities γ e ensure the existence of a factor H. Let G[H] be a blow-up of the graph H such that the density between A i A j is at least γ ij, where A i is a cluster of the vertex i V(H). We show that it contains a factor H. Let us consider a graph H U n,3 with n > 3 vertices. Let v,u V(H), where v is a leaf of H u N H (v). Define R v,u as the subset of A u in the following way (see Fig. 6). R v,u = {x A u x is incident to some edge between Note that Hence A u A v }. R v,u A v e(r v,u,a v ) = γ uv A u A v. R v,u γ uv A u. Now we show the lower bound for the number of edges incident to R v,u. Let k N H (u). By the inclusion - exclusion Fig. 6. Clusters A v A u with the set R v,u. formula we count the lower bound for the number of edges between R v,u A k as follows. e(r v,u,a k ) e(a u,a k ) ( A u R v,u ) A k = R v,u A k +(γ ku ) A k A u R v,u A k +(γ ku ) R v,u A k = γ uv ( r ) ku R v,u A k = γ ku r R v,u A k. uv Now, by deleting the vertex set A v A u \R v,u from G[H], we obtain a graph which is a blow-up of H with edge densities ensuring the existence of the factor H.

5 KAMIL POWROŹNIK, HALINA BIELAK: AN EFFICIENT ALGORITHM FOR THE DENSITY 483 Fig. 7. Clusters A a, A b A c with the sets R a,b R a,c. Fig. 8. We assume that G [H ] is without transversal H. The construction of the blow-up graph G[H] without transversal H for the case where v is a leaf in H H = H v. The cluster A u is in G [H ]. Let A u = A u u A v = {v} be clusters in G[H]. Moreover, by the definition of R v,u the factor H can be extended to a factor H. Now consider the situation when n = 3 graph H is a cycle C 3 with the vertex set {a,b,c}. Let A a be a cluster of vertex a. Define sets R a,b R a,c in the following way (see Fig. 7). R a,b = {x A b x is incident to some edge between A b A a }, R a,c = {x A c x is incident to some edge between Note that A c A a }. R b A a e(r b,a a ) = γ ab A a A b, R c A a e(r c,a a ) = γ ac A a A c. Hence we have the following lower bounds for the cardinalities of R a,b R a,c R a,b γ ab A b R a,c γ ac A c. Next we show how many edges are incident to R a,b R a,c. Using the inclusion - exclusion formula we count the lower bound for the number of edges between R a,b R a,c e(r a,b,r a,c ) e(a b,a a ) ( A b R a,b ) A c ( A c R a,c ) A b +( A b R a,b )( A c R a,c ) = R a,b R a,c +(γ bc ) A b A a R a,b R a,c +(γ bc ) R a,b R a,c = γ ab γ ac ( ) r bc R a,b R a,c = γ bc ( r ab )( r ac ) R a,b R a,c. Now, by deleting the vertex sets A a, A b \R a,b A c \R a,c fromg[c 3 ], we obtain a graph which is a blow-up ofc 3 = P 2, V(P 2 ) = {b,c}, with edge densities ensuring the existence of Fig. 9. We assume that G [H ] is without transversal H. The construction of the blow-up graph G[H] without transversal H for the case where c is a vertex of C 3, V(C 3 ) = {a,b,c} in H H = H c. The clusters A a A b are in G [H ]. Let A a = {a } A a, A b = {b } A b A c = {c} be clusters in G[H]. the factor P 2. Moreover, by the definition of R a,b R a,c the factor P 2 can be extended to a factor C 3. Note that if γ ku < 0, then γ ku +γ uv <. So there exists a construction which does not induce the path P 3 with the consecutive vertices k,u,v, where i A i (i {k,u,v}) in this case. Therefore, if some γ ku < 0 then there exists a construction for a blow-up graph ofh without a factor of H. Next assume that all the γ e are proper densities, but there is a construction of a blow-up graph, say G [H ], with edge densities at least γ e, but which does not induce a factor H. Thus we construct a blow-up G[H] of the graph H not inducing H. We consider two possible cases. First let the picked vertex v be a leaf in H H = H v. Then set A u = {u } A u A v = {v}. We connect v to all elements of A u but do not connect to u without changing densities in G [H ] with density γ vu (see Fig. 8). Now let H = H c, where c is a vertex of C 3, V(C 3 ) = {a,b,c}. Then set A a = {a } A a, A b = {b } A b A c = {c}. We connect c to all elements of A a A b but do not connect to a b without changing densities in G [H ] with densities γ ca γ cb (see Fig. 9).

6 484 PROCEEDINGS OF THE FEDCSIS. WARSAW, 204 Theorem 7 provides us with the algorithm (AlgorithmU n,3 ) to decide whether a given set of edge densities ensures the existence of a transversal H in a blow-up graph G[H] or does not ensure, where H U n,3. For further considerations recall some results presented in papers [3] [5]. First lemma gives condition on edge densities in the triangle C 3 which allows us to check if these densities ensure existing of C 3 in a blow-up graph G[C 3 ]. The second results gives condition on existing graph H as a factor in a blow-up graphg[h] in terms of the multivariate matching polynomial F H. Lemma. (Bondy, et al. [3]) Let α, β, γ be the edge densities between the clusters of a blow-up graph of the triangle - a cycle C 3. If αβ +γ >,βγ +α >,γα+β >, then the blow-up graph contains a triangle as a transversal. Theorem 8. (Csikvári, Nagy [5]) Assume that for the graph H we have F H (r e,t) > 0 for all t [0,] some vector r e of weights, where r e [0,] for each edge e E(H). Then the densities γ e = r e ensure the existence H as a transversal. Let H := C 3 with vertices a,b,c edge densities γ e = r e, where e {ab,bc,ac}. Assume that all r e [0,) run the Algorithm U n,3 by deleting vertex a from the graph C 3 with edges incident to it (means ab ac). As a result we get a graph H as a path P 2 = bc with edge density γ bc = r bc = r bc ( r ab )( r ac ). For H we have By Theorem 8 we need for t [0,]. Hence F H (r e,t) = tr bc. F H (r e,t) > 0 r bc >, ( r ab )( r ac ) r bc > 0 γ ab γ ac +γ bc >. Similar inequalities are received when, instead of a vertex a, we delete in Algorithm U n,3 vertex b or c. As we can see we have a result presented in Proposition consensual with Lemma. Proposition. Let a,b,c be vertices in a triangle C 3. Assume that γ e = r e be an edge density assigned to each edge e E(C 3 ), where E(C 3 ) = {ab,ac,bc}. If r ab ( r ac )( r bc ) <, r ac ( r ab )( r bc ) < r bc ( r ab )( r ac ) <, then the set of densities {γ e } e E(C3) ensures existence of a transversal C 3 in a blow-up graph G[C 3 ]. By running Algorithm U n,3 on some unicyclic graph H U n,3 with γ e = tr e using the multivariate matching polynomial F H (r e,s) we can prove the following lemma. Lemma 2. Let H be a weighted unicyclic graph of order n > 2 with the cyclec 3. Let γ e = tr e be densities assigned to each edge e E(H), where r e [0,). Assume that after running Algorithm U n,3 we get a cycle C 3 with F C3 (r e,t) = 0, then t is a root of the multivariate matching polynomial F H (r e,s) of the graph H. Proposition 2. Let H be a weighted unicyclic graph of order n > 2 with the cycle C 3. Let γ e = tr e be the densities assigned to each edge e E(H). Assume that after running Algorithm U n,3 we get a cycle C 3 with the vertex set V(C 3 ) = {a,b,c} with F C3 (r e,t) = 0, after restart Algorithm U n,3, we get a path P 2 (by deleting the vertex a edges ab, ac), then tr bc F P2 (r e,s) = t2 r ab r ac +tr bc ( tr ab )( tr ac ) s ( tr ab )( tr ac ). Proof. Assume that after running Algorithm U n,3 we get a cycle C 3 with edge densities γ e = tr e. Let V(C 3 ) = {a,b,c} r ab,r ac,r bc [0,). The multivariate matching polynomial has exactly one root F C3 (r e,s) = s(r ab +r ac +r bc ) t = (r ab +r ac +r bc ). By deleting vertex a from the cycle C 3 with the edges e ab = ab e ac = ac we obtain a path P 2 = bc. By Theorem 7 we get that tr bc F P2 (r e,s) = sr bc = s ( tr ab )( tr ac ). By multiplying both sides by we have So ( tr ab )( tr ac ) ( tr ab )( tr ac )F P2 (r e,s) = ( tr ab )( tr ac ) str bc = t(r ab +r ac +r bc )+tr b c+t 2 r ab r ac str bc. ( tr ab )( tr ac )F P2 (r e,s) t 2 r ab r ac tr bc +str bc = F C3 (r e,t) = 0.

7 KAMIL POWROŹNIK, HALINA BIELAK: AN EFFICIENT ALGORITHM FOR THE DENSITY 485 Algorithm U n,3 Input: a unicyclic graph H U n,3 with the set of edge densities {γ e } e E(H). Output: a boolean value TRUE, the densities γ e ensure the existence of a factor H, D = FALSE, the densities γ e does not ensure the existence of a factor H. Consider a weighted graph (H,r e ), where r e = γ e. Step. if V(H) = 2 (means H is a path P 2 ) 0 r e < then STOP: D :=TRUE. if V(H) 2 there exists an edge for which r e then STOP: D :=FALSE. Step 2. if V(H) = 3 (means H is a cycle C 3 ) 0 r e < for all edges e E(H) then pick a vertex c of the graph H let a,b be its neighbours. Let H := H c r ab = r ab ( r ac )( r bc ). if V(H) > 3 0 r e < for all edges e E(H) then pick a vertex v of degree let u be its unique neighbour. Let H := H v r e, if e is not incident to u, r e = r e r uv, if e is incident to u. Go to Step with (H,r e ) := (H,r e ). Hence tr bc F P2 (r e,s) = t2 r ab r ac +tr bc ( tr ab )( tr ac ) s ( tr ab )( tr ac ). By the definition of F P2 (r e,s) we have t 2 r ab r ac +tr bc ( tr ab )( tr ac ) = t(r ac +r ab +r bc ) =. then Therefore, tr bc = ( tr ab )( tr ac ) t 2 r ab r ac ( tr ab )( tr ac ) = 0, tr ab tr ac = 0. Note that if γ bc = tr bc ( tr ab )( tr ac ) = 0, t(r ac +r ab +r bc ) = +tr ab tr ac =. So t is the root of F C3 (r e,t).

8 486 PROCEEDINGS OF THE FEDCSIS. WARSAW, 204 From above consideration we deduce that Algorithm U n,3 works correctly with time complexity at most O(n 2 ). Algorithm U n,3 can be implemented in such a way that a vertex of the subgraph C 3 be considered (picked) in the last step of the algorithm. IV. CONCLUSION We have presented Algorithm U n,3 for testing whether the unicyclic graph H U n,3 with the set of edge densities {γ e } e E(H) is a factor of a blow-up graph G[H]. Precisely, we have the answer whether the edge densities ensure the existence of the factor or do not ensure. In future work we will study the density Turán problem for an arbitrary graph of the family U n,p, p 4, for other families of graphs. Moreover, we wish to construct efficient algorithms for testing the existence of blow-up graphs with factors of the families. Open problem: Look for the density Turán problem algorithm for families of connected graphs with blocks (i.e., 2-connected components) isomorphic to cycles /or P 2. REFERENCES [] R. Baber, J.R. Johnson J. Talbot, The minimal density of triangles in tripartite graphs, LMS J. Comput. Math., 3 (200), , [2] B. Bollobás, Extremal Graph Theory, Academic Press (978). [3] A. Bondy, J. Shen, S. Thomassé C. Thomassen, Density Conditions for triangles in multipartite graphs, Combinatorica, 26 (2006), [4] W.G Brown, P. Erdös M. Simonovits, Extremal problems for directed graphs, Journal of Combinatorial Theory, Series B 5 () (973), 77 93, [5] P. Csikvári Z. L. Nagy, The density Turán Problem, Combinatorics, Probability Computing, 2 (202), , [6] Z. Füredi, Turán type problems, Survey in Combinatorics Vol. 66 of London Math. Soc. Lecture Notes (A.D. Keedwell, ed.) (99), , [7] C.D. Godsil G. Royle, Algebraic Graph Theory, Springer (200), [8] G. Jin, Complete subgraphs of r-partite graphs, Combin. Probab. Comput., (992), , [9] Z.L. Nagy, A multipartite version of the Turán problem - density conditions eigenvalues, The Electronic Journal of Combinatorics, 8 (20), # P46. [0] P. Turán, On an extremal problem in graph theory, Mat. Fiz. Lapok, 48 (94), [] R. Yuster, Independent transversal in r-partite graphs, Discrete Math., 76 (997), ,