1,443 research outputs found

    Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers

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    AbstractThe notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition of a multipartite Ramsey number is broadened still further, by incorporating off-diagonal numbers, fixing the number of vertices per partite set in the larger graph and then seeking the minimum number of such partite sets that would ensure the occurrence of certain specified monochromatic multipartite subgraphs

    Graph Saturation in Multipartite Graphs

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    Let GG be a fixed graph and let F{\mathcal F} be a family of graphs. A subgraph JJ of GG is F{\mathcal F}-saturated if no member of F{\mathcal F} is a subgraph of JJ, but for any edge ee in E(G)−E(J)E(G)-E(J), some element of F{\mathcal F} is a subgraph of J+eJ+e. We let ex(F,G)\text{ex}({\mathcal F},G) and sat(F,G)\text{sat}({\mathcal F},G) denote the maximum and minimum size of an F{\mathcal F}-saturated subgraph of GG, respectively. If no element of F{\mathcal F} is a subgraph of GG, then sat(F,G)=ex(F,G)=∣E(G)∣\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|. In this paper, for k≥3k\ge 3 and n≥100n\ge 100 we determine sat(K3,Kkn)\text{sat}(K_3,K_k^n), where KknK_k^n is the complete balanced kk-partite graph with partite sets of size nn. We also give several families of constructions of KtK_t-saturated subgraphs of KknK_k^n for t≥4t\ge 4. Our results and constructions provide an informative contrast to recent results on the edge-density version of ex(Kt,Kkn)\text{ex}(K_t,K_k^n) from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure

    Transversal designs and induced decompositions of graphs

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    We prove that for every complete multipartite graph FF there exist very dense graphs GnG_n on nn vertices, namely with as many as (n2)−cn{n\choose 2}-cn edges for all nn, for some constant c=c(F)c=c(F), such that GnG_n can be decomposed into edge-disjoint induced subgraphs isomorphic to~FF. This result identifies and structurally explains a gap between the growth rates O(n)O(n) and Ω(n3/2)\Omega(n^{3/2}) on the minimum number of non-edges in graphs admitting an induced FF-decomposition

    A Short Note on Undirected Fitch Graphs

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    The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information

    Induced minors and well-quasi-ordering

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    A graph HH is an induced minor of a graph GG if it can be obtained from an induced subgraph of GG by contracting edges. Otherwise, GG is said to be HH-induced minor-free. Robin Thomas showed that K4K_4-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without K4K_4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for HH-induced minor-free graphs and show that the class of HH-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if HH is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]

    Finite 33-connected homogeneous graphs

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    A finite graph \G is said to be {\em (G,3)(G,3)-((connected)) homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most 33 extends to an automorphism g∈Gg\in G of the graph, where GG is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite (G,3)(G, 3)-homogeneous graphs. In this paper, we develop a method for characterising (G,3)(G,3)-connected homogeneous graphs. It is shown that for a finite (G,3)(G,3)-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is 22--transitive or G_v^{\G(v)} is of rank 33 and \G has girth 33, and that the class of finite (G,3)(G,3)-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where GG is quasiprimitive on VV. We determine the possible quasiprimitive types for GG in this case and give new constructions of examples for some possible types
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