LIST COLORING OF COMPLETE MULTIPARTITE GRAPHS

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1 LIST COLORING OF COMPLETE MULTIPARTITE GRAPHS Tomáš Vetrík School of Mathematical Sciences University of KwaZulu-Natal Durban, South Africa Abstract The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each verte v of G, there is a proper coloring of G that assigns to each verte v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r partite classes of order two. Keywords: list coloring, choice number, complete multipartite graph. 000 Mathematics Subject Classification: 05C5.. Introduction All graphs considered here are finite, undirected, without loops and multiple edges. Let G be a graph with the verte set V (G) and the edge set E(G). A list assignment to the vertices of a graph G is the assignment of a list L(v) of colors C to every verte v V (G). A k-list assignment is a list assignment such that L(v) k for every verte v. An L-coloring of G is a function f : V (G) C such that f(v) L(v) for all v V (G) and f(v) f(w) for each edge vw E(G). If G has an L-coloring, then G is said to be L-colorable. If for any k-list assignment L there eists an L-coloring, then G is k-choosable. The choice number Ch(G) of a graph G is the minimum integer k such that G is k-choosable. The study of choice numbers of graphs was initiated by Vizing [7] and by Erdős, Rubin and Taylor [3]. For a survey about the list coloring problem we refer to [6] and [8]. In this paper we focus on the choice numbers of complete multipartite graphs.

2 . Complete multipartite graphs with partite classes of different sizes Let K n,n,...,n r be the complete r-partite graph with the partite classes of order n, n,..., n r. A well-known result of Erdős, Rubin and Taylor [3] says that the choice number of the complete r-partite graph K,,..., is r. Gravier and Maffray [4] proved that also Ch(K 3,3,,..., ) = r for r 3. Enomoto et. al. [] showed that Ch(K 5,,..., ) = r + and the choice number of the complete r-partite graph K 4,,..., is equal to r if r is odd, and r + if r is even. Motivated by these results we study the value Ch(K n,,..., ) for any positive integer n. In the proof of Theorem we write L(S) for the union v S L(v) where S V (G). If C is a set of colors, then L\C denotes the list assignment obtained from L by removing the colors in C from each L(v) where v V (G). First, we show that the graph K (t+)(t+3)/,,..., is (r + t)- choosable. Theorem. Let t be a positive integer and let G be a complete r-partite graph with one partite class of order (t + )(t + 3)/ and r partite classes of order two. Then Ch(G) r + t. Proof. Let V be the partite class of G of order (t + )(t + 3)/ and let V i = {v i, w i }, i r, be the partite classes of order two. Let L be any (r + t)-list assignment to the vertices of G. We prove that G is L -colorable. We distinguish three cases: CASE : t r. We can color the vertices of V, V 3,..., V r with r different colors. Since L (v) r for every verte v V, we can color the vertices of V as well. CASE : There eists a color c L (v i ) L (w i ) for some i {, 3,..., r}. It is easy to show by induction on r that G is L -colorable. The step r = is trivial. For the induction step, assign c to both v i and w i, and remove c from the lists of the remaining vertices. By the induction hypothesis, the remaining vertices can be colored with colors from the revised lists. CASE 3: t r and L (v i ) L (w i ) = for every i {, 3,..., r}. We prove by contradiction that G is L -colorable. Assume that G is not L -colorable. Let L be an (r + t)-list assignment such that G is not L- colorable. Let X j, j =,,..., t, be the largest subset of V \( j l= X l) with

3 v X j L(v). Set X j = j and choose a color c j v X j L(v). Define L = L\{c, c,..., c t } and G = G\( t l= X l). Note that L (v) = r + t for each v V (G ) V and L (v i ), L (w i ) r for any i {, 3,..., r}. Since G is not L-colorable, G is not L -colorable. It follows that there eists a set of vertices T V (G ) such that L (T ) < T, i.e., L does not satisfy Hall s condition. Let S denote a maimal subset of V (G ) such that L (S) < S. We consider two subcases: CASE 3a: S V i for every i {, 3,..., r}. Since L (v i ), L (w i ) r and S\V r, S\V can be colored from the list L. Further, L (v) = r + t for v S V, therefore we can also color the vertices in S V. Let L = L \L (S). We show that G \S is L -colorable. If G \S is not L -colorable, we have a nonempty subset S V (G )\S with L (S ) < S. Then L (S S ) = L (S) + L (S ) < S + S, which contradicts the maimality of S. CASE 3b: Both v i, w i S for some i {, 3,..., r}. Then S > L (S) L (v i ) + L (w i ) (r + t) t. Set S = r + t + + ɛ where ɛ 0. Clearly, L (S) r + t + ɛ. Let S = S V. We have S S (r ) = t ɛ. By the maimality of X t, every color in L (S) appears in the lists of at most t vertices of S. It means that (r + t) S = v S L (v) t L (S). () It is evident that t l= l + S V = (t + )(t + 3)/. Hence, t t + S (t + )(t + 3)/, or equivalently t [(t + )(t + 3)/ S ]/t () By () and (), we have (r + t) S [(t + )(t + 3)/ S ] L (S) /t. Since S t+3+ɛ and L (S) r +t+ɛ, we have (r +t)(t+3+ɛ) [(t+)(t+ 3)/ (t+3+ɛ)](r+t+ɛ)/t which yields t3 +(3+ɛ) t +(r )ɛt+(r+ɛ)ɛ 0, a contradiction. This finishes the proof. If t =, then Ch(K 6,,..., ) r+. This bound also comes from the result Ch(K 3,3,,..., ) = r of Gravier and Maffray [4], because the complete r-partite graph K 6,,..., is a subgraph of the complete (r + )-partite graph K 3,3,,...,. Since the choice number of the complete r-partite graph K 5,,..., is equal to r +, it is clear that Ch(K 6,,..., ) = r + as well. 3

4 Now we present a lower bound on the choice number of complete r-partite graphs with r partite classes of order at most two. Theorem. Let s, r, t be integers such that 0 s < r and t > 0. Let G be a complete r-partite graph consisting of one partite class of order ( ) t+s, t r s partite classes of order two, and s partite classes of order one. Then Ch(G) > r+t (t + s). t+s Proof. Let n = ( ) t+s t and m = r+t. Let G be a complete r-partite t+s graph with the partite classes V, V i = {v i, w i }, V j = {v j }, where V = n; i =, 3,..., r s and j = r s+, r s+,..., r. Let A, A,..., A t+s, B, B,..., B t+s be disjoint color sets of order m such that t+s i= A i = A, t+s i= B i = B. We define a list assignment L to V (G) by the following way: L(v j ) = A, j =, 3,..., r, L(w i ) = B, i =, 3,..., r s. The lists of colors given to the vertices of V consist of t + s different sets A, A,..., A t+s, B y, B y,..., B yt, where,,..., t+s, y, y,..., y t )( t+s t+s t {,,... t + s}. Since the number of vertices in V is n = ( ) t+s, we are able to assign to any two vertices in V different lists. We show by contradiction that G can not be colored from the list L. Suppose that G can be colored from L. We use r different colors of A to color the vertices v, v 3,..., v r and r s different colors of B to color w, w 3,..., w r s. Since A = B = m (t + s) r + t, the number of colors in A (in B) not used to color V, V 3,..., V r is at most t (at most t+s). It follows that there are at most t+s sets A, A,..., A t, B y, B y,..., B y t+s, where,,..., t, y, y,..., y t+s {,,... t + s} containing colors that were not employed in coloring V, V 3,..., V r. Try to color V with these colors. According to the assignment of color sets to the vertices of V, there eists a verte v V having none of the sets A, A,..., A t, B y, B y,..., B y t+s in its list, a contradiction. Hence, G is not L-colorable. Note that we get the bound Ch(K ( t t ) ) r+t if s = 0 and r = pt+,,..., for some odd integer p. 3. Complete multipartite graphs with partite classes of equal sizes 4

5 Let K n r denote the complete multipartite graph with r partite classes of order n. The problem is to determine the value of the choice number Ch(K n r ). If n =, then K n r is a clique on r vertices and hence, obviously, Ch(K r ) = r. In the previous section we mentioned that Ch(K r ) = r as well. Alon [] established the general bounds c r log n Ch(K n r ) c r log n for every r, n, where c, c are two positive constants. Later, Kierstead [5] solved the problem in the case n = 3. He showed that Ch(K 3 r ) = 4r. 3 Yang [9] studied the value of Ch(K 4 r ) and obtained the bounds 3r Ch(K 4 r ) 7r. We present results giving eact bounds on Ch(K 4 n r) for large n. In the proof of Theorem 3 we use the following lemma proved in [5]. Lemma. A graph G is k-choosable if G is L-colorable for every k-list assignment L such that v V (G) L(v) < V (G). Let us derive an upper bound on the choice number of complete multipartite graphs with partite classes of equal sizes. Theorem 3. Let 0 < α < n and let j = (α α j n l= l) +, j =,,..., α. If n α l= l, then Ch(K n r ) αr. Proof. Let V i, i =,,..., r, be the i-th partite class of K n r. We prove the result by induction on r. The case r = is trivial. For the induction step consider an αr -list assignment L to the vertices of K n r. We prove that if n α l= l, then any partite class V i can be colored with at most α colors. Assume that n = α l= l. In this paragraph we show by induction on j (j =,,..., α ), that there eists a color c j which can be used for coloring j vertices of V i that have not been colored by c, c,..., c j yet. Note that c l, c l, where l, l {,,..., α }, l l, do not have to be different. If j =, we have = α +. Since v V i L(v) = αr n and by Lemma, v V (K L(v) < rn, there eists a color c n r) which appears in the lists of at least α + vertices of V i. Color these vertices with c. Suppose j. We can color j l= l vertices with c, c,..., c j. The sum of the numbers of colors in the lists of the remaining n j l= l vertices of V i is (n j l= l) αr. Since v V i L(v) < rn, there is a color c j that appears in the lists of other (n j l= l) α + = n j vertices. Hence, we can color these vertices with c j. vertices of V i with at most α different colors. It follows that it is possible to color n = α l= l 5

6 Clearly, if n < α l= l, all the vertices of V i can be colored with at most α colors too. Let us remove the colors that were employed in coloring V i from the lists given to the vertices in V (K n r )\V i. We have at least αr α colors. Since αr α α(r ), by applying the induction hypothesis, r partite classes can be colored with α(r ) colors, i.e., there eists a proper coloring of the vertices in V (K n r )\V i with colors from the revised lists. Unfortunately, the result presented in Theorem 3 can not be bounded from above by cr log n, where c is a constant. Theorem 3, for eample, yields the upper bounds Ch(K 5 r ) 5 r, Ch(K 5 r) 5r, Ch(K 40 r ) 0r, Ch(K 75 r ) 5r and Ch(K r ) 0r. One can check that 0r 6.4r log 40, 5r 8r log 75 and 0r 9.6r log. The following result gives a lower bound on Ch(K n r ). Theorem ( 4. Let, t, r, n be integers such that, t, r, t and n = ) t+. Then Ch(Kn r ) > ( t + ) tr. Proof. Let, t, r, t, n = ( ) t+ and let k = ( t + ) tr. We show that there eists a k-list assignment L of K n r such that K n r is not L-colorable. Let V i, i =,,..., r, be the i-th partite class of K n r. Let A, A,..., A be a family of disjoint color sets such that A j = A or A j = A +, j =, 3,...,, and j= A j = tr. Obviously, A j tr for any j {,,..., }. Define a list assignment L as follows: Let the lists given to the n vertices of every partite class V i consist of t + different sets A y, A y,..., A y t+, y, y,..., y t+ {,,..., }, where any two vertices in the same part have different lists. Note that L(v) ( t + ) tr for each verte v V (K n r ). Then for any partite class V i and any t colors a j A y j, j =,,..., t ; y j {,,..., } there is a verte v V i having none of the sets A y j in its list. Therefore, in any coloring from these lists, we must use at least t colors on each partite class. Since the number of colors in j= A j is less than tr, K n r is not L-colorable. Theorem 4 says that if, for instance t =, then n = and Ch(K n r ) > (n ) r. In particular, for n = 5 we have Ch(K n 5 r) > 4 r. If 5 t = 3, then Ch(K n r ) > ( ) 3r. For eample, in the case = 6 we get Ch(K 5 r ) > 4 3r = 4 r. 6 Finally, we present a corollary of Theorem 4 which yields a lower bound 6

7 in the form cr log n. Corollary. Let r and n = ( / ) where 5. Then Ch(Kn r ) > r log. n. Proof. For, t, r, t and n = ( t+), we have Ch(Kn r ) > ( t + ) tr. Let t = +. Then Ch(K n r) > / r+r r. It is well-known that! (+)+ for any. For 5, the following inequalities also hold: <! < 6+! e e. Then n = < e 5e /! /! 6 + /(5e ) Since < 7.6(.) for any 4 / / / / /e 0 / e 0( ) e (note that 7.5(.) < for 9 ) and ( 5, we have n < 7.068(.) e Ch(K n r ) > r log. n for any n = ( ) < 3. for any < (.). Consequently, log. n <, hence / ) where 5. References [] N. Alon, Choice numbers of graphs; a probabilistic approach, Combinatorics, Probability and Computing (99), [] H. Enomoto, K. Ohba, K. Ota and J. Sakamoto, Choice number of some complete multi-partite graphs, Discrete Mathematics 44 (00), [3] P. Erdős, A. L. Rubin and H. Taylor, Choosability in graphs, Proceedings of the West-Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, California, Congressus Numerantium XXVI (979), [4] S. Gravier and F. Maffray, Graphs whose choice number is equal to their chromatic number, Journal of Graph Theory 7 (998), [5] H. A. Kierstead, On the choosability of complete multipartite graphs with part size three, Discrete Mathematics (000), [6] Zs. Tuza, Graph colorings with local constraints a survey, Discussiones Mathematicae Graph Theory 7 (997), 6 8. [7] V. G. Vizing, Coloring the vertices of a graph in prescribed colors, Diskret. Analiz 9 (976), 3 0 (in Russian). 7

8 [8] D. R. Woodall, List colourings of graphs. Surveys in Combinatorics, London Mathematical Society Lecture Note Series 88, Cambridge University Press (00), [9] D. Yang, Etension of the game coloring number and some results on the choosability of complete multipartite graphs, PhD Thesis, Arizona State University (003), 8 pages. 8