70 research outputs found

    Countable graphs are majority 3-choosable

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    The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a 3-colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this conjecture, and proved that such a colouring exists for lists of size 4. We improve their result to lists of size 3; the proof extends to directed acyclic graphs. We also discuss some generalisations.Comment: 6 pages. Minor changes including adding a referenc

    On generalised majority edge-colourings of graphs

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    A 1k\frac{1}{k}-majority ll-edge-colouring of a graph GG is a colouring of its edges with ll colours such that for every colour ii and each vertex vv of GG, at most 1k\frac{1}{k}'th of the edges incident with vv have colour ii. We conjecture that for every integer k2k\geq 2, each graph with minimum degree δk2\delta\geq k^2 is 1k\frac{1}{k}-majority (k+1)(k+1)-edge-colourable and observe that such result would be best possible. This was already known to hold for k=2k=2. We support the conjecture by proving it with 2k22k^2 instead of k2k^2, which confirms the right order of magnitude of the conjectured optimal lower bound for δ\delta. We at the same time improve the previously known bound of order k3logkk^3\log k, based on a straightforward probabilistic approach. As this technique seems not applicable towards any further improvement, we use a more direct non-random approach. We also strengthen our result, in particular substituting 2k22k^2 by (74+o(1))k2(\frac{7}{4}+o(1))k^2. Finally, we provide the proof of the conjecture itself for k4k\leq 4 and completely solve an analogous problem for the family of bipartite graphs.Comment: 18 page

    Countable graphs are majority 3-choosable

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    The Unfriendly Partition Conjecture posits that every countable graph admits a -colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a -colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this conjecture, and proved that such a colouring exists for lists of size . We improve their result to lists of size ; the proof extends to directed acyclic graphs. We also discuss some generalisations

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    Graph Relations and Constrained Homomorphism Partial Orders

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    We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every d3d\geq 3 the homomorphism order on the class of line graphs of graphs with maximum degree dd is universal

    Homomorphism complexes, reconfiguration, and homotopy for directed graphs

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    The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs GG and HH, we consider the polyhedral complex Hom(G,H)\text{Hom}(G,H) that parametrizes the directed graph homomorphisms f:GHf: G \rightarrow H. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals. We study examples of directed Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom(G,Tn)\text{Hom}(G,T_n) for TnT_n a tournament. Finally, we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified statements and proofs, other corrections and minor revisions incorporating comments from referee

    Master index of volumes 61–70

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    Master index: volumes 31–40

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    Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey

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    This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter
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