36 research outputs found

    Maximal induced matchings in triangle-free graphs

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    An induced matching in a graph is a set of edges whose endpoints induce a 11-regular subgraph. It is known that any nn-vertex graph has at most 10n/5≈1.5849n10^{n/5} \approx 1.5849^n maximal induced matchings, and this bound is best possible. We prove that any nn-vertex triangle-free graph has at most 3n/3≈1.4423n3^{n/3} \approx 1.4423^n maximal induced matchings, and this bound is attained by any disjoint union of copies of the complete bipartite graph K3,3K_{3,3}. Our result implies that all maximal induced matchings in an nn-vertex triangle-free graph can be listed in time O(1.4423n)O(1.4423^n), yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.Comment: 17 page

    On the multiple Borsuk numbers of sets

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    The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

    Open problems on graph coloring for special graph classes.

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    For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring
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