121 research outputs found

    Subdivisions of a large clique in C6-free graphs

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    Mader conjectured that every -free graph has a subdivision of a clique of order linear in its average degree. We show that every -free graph has such a subdivision of a large clique. We also prove the dense case of Mader's conjecture in a stronger sense, i.e., for every c, there is a such that every -free graph with average degree has a subdivision of a clique with where every edge is subdivided exactly 3 times

    A proof of Mader's conjecture on large clique subdivisions in C4C_4-free graphs

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    Given any integers s,t2s,t\geq 2, we show there exists some c=c(s,t)>0c=c(s,t)>0 such that any Ks,tK_{s,t}-free graph with average degree dd contains a subdivision of a clique with at least cd12ss1cd^{\frac{1}{2}\frac{s}{s-1}} vertices. In particular, when s=2s=2 this resolves in a strong sense the conjecture of Mader in 1999 that every C4C_4-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of Ks,tK_{s,t}-free graphs suggests our result is tight up to the constant c(s,t)c(s,t).Comment: 25 pages, 1 figur

    Clique complexes and graph powers

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    We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.Comment: V3: final versio

    Maximum Independent Sets in Subcubic Graphs: New Results

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    The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known results

    Vertex elimination orderings for hereditary graph classes

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    We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

    Rainbow clique subdivisions

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    We show that for any integer t2t \ge 2, every properly edge colored nn-vertex graph with average degree at least (logn)2+o(1)(\log n)^{2+o(1)} contains a rainbow subdivision of a complete graph of size tt. Note that this bound is within a log factor of the lower bound. This also implies a result on the rainbow Tur\'{a}n number of cycles
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