349 research outputs found

    All partitions have small parts - Gallai-Ramsey numbers of bipartite graphs

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    Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it suffices to consider only 33-colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part and initiate the study of linear forests

    The typical structure of Gallai colorings and their extremal graphs

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    An edge coloring of a graph GG is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai rr-colorings of KnK_n is ((r2)+o(1))2(n2)\left(\binom{r}{2}+o(1)\right)2^{\binom{n}{2}}. This result indicates that almost all Gallai rr-colorings of KnK_n use only 2 colors. We also study the extremal behavior of Gallai rr-colorings among all nn-vertex graphs. We prove that the complete graph KnK_n admits the largest number of Gallai 33-colorings among all nn-vertex graphs when nn is sufficiently large, while for r4r\geq 4, it is the complete bipartite graph Kn/2,n/2K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}. Our main approach is based on the hypergraph container method, developed independently by Balogh, Morris, and Samotij as well as by Saxton and Thomason, together with some stability results for containers.Comment: 28 page

    Brooks's theorem for measurable colorings

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    We generalize Brooks's theorem to show that if GG is a Borel graph on a standard Borel space XX of degree bounded by d3d \geq 3 which contains no (d+1)(d+1)-cliques, then GG admits a μ\mu-measurable dd-coloring with respect to any Borel probability measure μ\mu on XX, and a Baire measurable dd-coloring with respect to any compatible Polish topology on XX. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID dd-colorings of Cayley graphs of degree dd, except in two exceptional cases.Comment: Minor correction

    A conjecture on Gallai-Ramsey numbers of even cycles and paths

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    A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai kk-coloring is a Gallai coloring that uses at most kk colors. Given an integer k1k\ge1 and graphs H1,,HkH_1, \ldots, H_k, the Gallai-Ramsey number GR(H1,,Hk)GR(H_1, \ldots, H_k) is the least integer nn such that every Gallai kk-coloring of the complete graph KnK_n contains a monochromatic copy of HiH_i in color ii for some i{1,2,,k}i \in \{1,2, \ldots, k\}. When H=H1==HkH = H_1 = \cdots = H_k, we simply write GRk(H)GR_k(H). We study Gallai-Ramsey numbers of even cycles and paths. For all n3n\ge3 and k2k\ge2, let Gi=P2i+3G_i=P_{2i+3} be a path on 2i+32i+3 vertices for all i{0,1,,n2}i\in\{0,1, \ldots, n-2\} and Gn1{C2n,P2n+1}G_{n-1}\in\{C_{2n}, P_{2n+1}\}. Let ij{0,1,,n1} i_j\in\{0,1,\ldots, n-1 \} for all j{1,2,,k}j\in\{1,2, \ldots, k\} with i1i2ik i_1\ge i_2\ge\cdots\ge i_k . The first author recently conjectured that GR(Gi1,Gi2,,Gik)=Gi1+j=2kij GR(G_{i_1}, G_{i_2}, \ldots, G_{i_k}) = |G_{i_1}|+\sum_{j=2}^k i_j. The truth of this conjecture implies that GRk(C2n)=GRk(P2n)=(n1)k+n+1GR_k(C_{2n})=GR_k(P_{2n})=(n-1)k+n+1 for all n3n\ge3 and k1k\ge1, and GRk(P2n+1)=(n1)k+n+2GR_k(P_{2n+1})=(n-1)k+n+2 for all n1n\ge1 and k1k\ge1. In this paper, we prove that the aforementioned conjecture holds for n{3,4}n\in\{3,4\} and all k2k\ge2. Our proof relies only on Gallai's result and the classical Ramsey numbers R(H1,H2)R(H_1, H_2), where H1,H2{C8,C6,P7,P5,P3}H_1, H_2\in\{C_8, C_6, P_7, P_5, P_3\}. We believe the recoloring method we developed here will be very useful for solving subsequent cases, and perhaps the conjecture

    List colorings of K5K_5-minor-free graphs with special list assignments

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    A {\it list assignment} LL of a graph GG is a function that assigns a set (list) L(v)L(v) of colors to every vertex vv of GG. Graph GG is called {\it LL-list colorable} if it admits a vertex coloring ϕ\phi such that ϕ(v)L(v)\phi(v)\in L(v) for all vV(G)v\in V(G) and ϕ(v)ϕ(w)\phi(v)\not=\phi(w) for all vwE(G)vw\in E(G). The following question was raised by Bruce Richter. Let GG be a planar, 3-connected graph that is not a complete graph. Denoting by d(v)d(v) the degree of vertex vv, is GG LL-list colorable for every list assignment LL with L(v)=min{d(v),6}|L(v)|=\min \{d(v), 6\} for all vV(G)v\in V(G)? More generally, we ask for which pairs (r,k)(r,k) the following question has an affirmative answer. Let rr and kk be integers and let GG be a K5K_5-minor-free rr-connected graph that is not a Gallai tree (i.e., at least one block of GG is neither a complete graph nor an odd cycle). Is GG LL-list colorable for every list assignment LL with L(v)=min{d(v),k}|L(v)|=\min\{d(v),k\} for all vV(G)v\in V(G)? We investigate this question by considering the components of G[Sk]G[S_k], where Sk:={vV(G)d(v)<k}S_k:=\{v\in V(G) | d(v)<k\} is the set of vertices with small degree in GG. We are especially interested in the minimum distance d(Sk)d(S_k) in GG between the components of G[Sk]G[S_k].Comment: 14 pages, 2 figures, 2 table

    Gallai-Ramsey number of an 8-cycle

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    Given graphs GG and HH and a positive integer kk, the Gallai-Ramsey number grk(G:H)gr_{k}(G : H) is the minimum integer NN such that for any integer nNn \geq N, every kk-edge-coloring of KnK_{n} contains either a rainbow copy of GG or a monochromatic copy of HH. These numbers have recently been studied for the case when G=K3G = K_{3}, where still only a few precise numbers are known for all kk. In this paper, we extend the known precise Gallai-Ramsey numbers to include H=C8H = C_{8} for all kk

    A Brooks type theorem for the maximum local edge connectivity

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    For a graph GG, let \cn(G) and \la(G) denote the chromatic number of GG and the maximum local edge connectivity of GG, respectively. A result of Dirac \cite{Dirac53} implies that every graph GG satisfies \cn(G)\leq \la(G)+1. In this paper we characterize the graphs GG for which \cn(G)=\la(G)+1. The case \la(G)=3 was already solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}. We show that a graph GG with \la(G)=k\geq 4 satisfies \cn(G)=k+1 if and only if GG contains a block which can be obtained from copies of Kk+1K_{k+1} by repeated applications of the Haj\'os join.Comment: 15 pages, 1 figur

    Multicolor Gallai-Ramsey numbers of C9C_9 and C11C_{11}

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    A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai kk-coloring is a Gallai coloring that uses kk colors. We study Ramsey-type problems in Gallai colorings. Given an integer k1k\ge1 and a graph HH, the Gallai-Ramsey number GRk(H)GR_k(H) is the least positive integer nn such that every Gallai kk-coloring of the complete graph on nn vertices contains a monochromatic copy of HH. It turns out that GRk(H)GR_k(H) is more well-behaved than the classical Ramsey number Rk(H)R_k(H). However, finding exact values of GRk(H)GR_k (H) is far from trivial. In this paper, we study Gallai-Ramsey numbers of odd cycles. We prove that for n{4,5}n\in\{4,5\} and all k1k\ge1, GRk(C2n+1)=n2k+1GR_k(C_{2n+1})= n\cdot 2^k+1. This new result provides partial evidence for the first two open cases of the Triple Odd Cycle Conjecture of Bondy and Erd\H{o}s from 1973. Our technique relies heavily on the structural result of Gallai on Gallai colorings of complete graphs. We believe the method we developed can be used to determine the exact values of GRk(C2n+1)GR_k(C_{2n+1}) for all n6n\ge6.Comment: A long and technical proof of Gallai-Ramsey numbers of C9 can be found in our preprint arXiv:1709.06130, which will not be submitted for publicatio

    Gallai-Ramsey numbers of C9C_9 with multiple colors

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    We study Ramsey-type problems in Gallai-colorings. Given a graph GG and an integer k1k\ge1, the Gallai-Ramsey number grk(K3,G)gr_k(K_3,G) is the least positive integer nn such that every kk-coloring of the edges of the complete graph on nn vertices contains either a rainbow triangle or a monochromatic copy of GG. It turns out that grk(K3,G)gr_k(K_3, G) behaves more nicely than the classical Ramsey number rk(G)r_k(G). However, finding exact values of grk(K3,G)gr_k (K_3, G) is far from trivial. In this paper, we prove that grk(K3,C9)=42k+1gr_k(K_3, C_9)= 4\cdot 2^k+1 for all k1k\ge1. This new result provides partial evidence for the first open case of the Triple Odd Cycle Conjecture of Bondy and Erd\H{o}s from 1973. Our technique relies heavily on the structural result of Gallai on edge-colorings of complete graphs without rainbow triangles. We believe the method we developed can be used to determine the exact values of grk(K3,Cn)gr_k(K_3, C_n) for odd integers n11n\ge11.Comment: 15 pages, 3 figures, one overlooked case, namely Claim 2.10, was adde

    Gallai colorings and domination in multipartite digraphs

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    Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set U=iSAiU=\cup_{i\in S} A_i is called a dominating set of size |S| if for any vertex viSAiv\in \cup_{i\notin S} A_i there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colorings, edge colorings of graphs without 3-colored triangles.Comment: 15 pages, 8 figure
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