148 research outputs found

    Stability for the Erdős-Rothschild problem

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    Given a sequence k:=(k1,,ks)\boldsymbol {k} := (k_1,\ldots ,k_s) of natural numbers and a graph G, let F(G;k)F(G;\boldsymbol {k}) denote the number of colourings of the edges of G with colours 1,,s1,\dots ,s , such that, for every c{1,,s}c \in \{1,\dots ,s\} , the edges of colour c contain no clique of order kck_c . Write F(n;k)F(n;\boldsymbol {k}) to denote the maximum of F(G;k)F(G;\boldsymbol {k}) over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/(n2)\log _2 F(n;\boldsymbol {k})/{n\choose 2} as n tends to infinity and proved a stability theorem for complete multipartite graphs G

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Topics in graph colouring

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    In this thesis, we study two variants of graph (vertex) colourings: multicolouring and correspondence colouring. In ordinary graph colouring, each vertex receives a colour. Such a colouring is proper if adjacent vertices receive different colours. In k-multi-colouring, each vertex receives a set of k colours, and such a multi-colouring is proper if adjacent vertices receive disjoint set of colours. A graph is (n, k)-colourable if there is a proper k-multi-colouring of it using n colours. In the first part of the thesis, we study the following two questions. 1. For given n, k and n′, k′, if a graph is (n, k)-colourable, then what is the largest subgraph of it that is (n′, k′)-colourable? 2. For given n, k, if a graph is (n, k)-colourable, then for what n′, k′ is the whole graph (n′, k′)-colourable? Question 1 is inspired by a partial colouring conjecture asked by Albertson, Grossman, and Haas [2] in 2000 regarding list colouring. We obtain exact answers for specific values of the parameters, and upper and lower bounds on the largest (n′, k′)-colourable subgraph for general values of n′, k′. For Question 2, we first observe how it can be reformulated into a conjecture by Stahl from 1976 regarding Kneser graphs, and prove new results towards Stahl’s conjecture. In the second part of the thesis, we study another variant of colouring, which is known as correspondence colouring. In correspondence colouring, each vertex is associated with a prespecified list of colours, and there is prespecified correspondence associated with each edge specifying which pair of colours from the two endvertices correspond. (On each edge, a colour on one endvertex corresponds to at most one colour on the other endvertex.) A correspondence colouring is proper if each vertex receives a colour from its prespecified list, and that for each edge, the colours on its endvertices do not correspond. A graph is n-correspondence-colourable if a proper correspondence colouring exist for any prespecified correspondences on any prespecified n-colour-lists associated to each vertex. As correspondence colouring is a generalisation of list colouring, it is natural to ask whether Albertson, Grossman, and Haas’ conjecture can be generalised to correspondence colouring. Unfortunately, there are graphs on which their conjectured value does not hold, and we will present a series of them. We then study: for given n and n′, how many vertices of a n-correspondence colourable graph can always be properly correspondence-coloured with arbitrary correspondences and arbitrary n′-colour-lists on that graph? We generalise some results from the original conjecture in list colouring. Then we discuss some sufficient conditions for a proper correspondence colouring to exist. The correspondence chromatic number of a graph is the smallest n such that the graph is n-correspondence colourable. We study how different graph operations affect the correspondence chromatic number of multigraphs, in which multiple edges are allowed

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    The List Square Coloring Conjecture fails for cubic bipartite graphs and planar line graphs

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    Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartie graphs and alternatively they developed their construction to bipartite graphs such that one partite set has maximum degree 77. Motivated by the List Total Coloring Conjecture, they also asked whether this number can be pushed down to 22. At about the same time, Kim, SooKwon, and Park (2015) asked whether there would exist a claw-free counterexample to establish a generalization for a conjecture of Gravier and Maffray (1997). In this note, we answer the problem of Kim and Park by pushing the desired upper bound down to 33 by introducing a family of cubic bipartite counterexamples, and positively answer the problem of Kim, SooKwon, and Park by introducing a family of planar line graphs

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Path and cycle decompositions of graphs and digraphs

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    In this thesis, we make progress on five long standing conjectures on path and cycle decompositions of graphs and digraphs. Firstly, we confirm a conjecture of Jackson from 1981 by showing that the edges of any sufficiently large regular bipartite tournament can be decomposed into Hamilton cycles. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs. Secondly, we determine the minimum number of paths required to decompose the edges of any sufficiently large tournament of even order, thus resolving a conjecture of Alspach, Mason, and Pullman from 1976. We also prove an asymptotically optimal result for tournaments of odd order. Finally, we give asymptotically best possible upper bounds on the minimum number of paths, cycles, and cycles and edges required to decompose the edges of any sufficiently large dense graph. This makes progress on three famous conjectures from the 1960s: Gallai's conjecture, Hajós' conjecture, and the Erdős-Gallai conjecture, respectively. This includes joint work with António Girão, Daniela Kühn, Allan Lo, and Deryk Osthus

    Ryser Type Conditions for Extending Colorings of Triples

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    In 1951, Ryser showed that an n×nn\times n array LL whose top left r×sr\times s subarray is filled with nn different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order nn if and only if the number of occurrences of each symbol in LL is at least r+snr+s-n. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let X,YX,Y be finite sets with XYX\subsetneq Y and Y0(mod3)|Y|\equiv 0 \pmod 3. When can we extend a (proper) coloring of λ(X3)\lambda \binom{X}{3} (all triples on a ground set XX, each one being repeated λ\lambda times) to a coloring of λ(Y3)\lambda \binom{Y}{3} using the fewest possible number of colors? It is necessary that the number of triples of each color in (X3)\binom{X}{3} is at least X2Y/3|X|-2|Y|/3. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483--495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using H\"aggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295--313), we show that the number of triples of each color being at least X/2Y/6|X|/2-|Y|/6 is sufficient. Finally we prove an Evans type result by showing that if Y3X|Y|\geq 3|X|, then any qq-coloring of any subset of λ(X3)\lambda \binom{X}{3} can be embedded into a λ(Y12)\lambda\binom{|Y|-1}{2}-coloring of λ(Y3)\lambda \binom{Y}{3} as long as qλ(Y12)λ(X3)/X/3q\leq \lambda \binom{|Y|-1}{2}-\lambda \binom{|X|}{3}/\lfloor{|X|/3}\rfloor.Comment: 10 page

    Advances in Discrete Applied Mathematics and Graph Theory

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    The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
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