34,987 research outputs found

    Coloring of two-step graphs: open packing partitioning of graphs

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    The two-step graphs are revisited by studying their chromatic numbers in this paper. We observe that the problem of coloring of two-step graphs is equivalent to the problem of vertex partitioning of graphs into open packing sets. With this remark in mind, it can be considered as the open version of the well-known 22-distance coloring problem as well as the dual version of total domatic problem. The minimum kk for which the two-step graph N(G)\mathcal{N}(G) of a graph GG admits a proper coloring assigning kk colors to the vertices is called the open packing partition number po(G)p_{o}(G) of GG, that is, p_{o}(G)=\chi\big{(}\mathcal{N}(G)\big{)}. We give some sharp lower and upper bounds on this parameter as well as its exact value when dealing with some families of graphs like trees. Relations between pop_{o} and some well-know graph parameters have been investigated in this paper. We study this vertex partitioning in the Cartesian, direct and lexicographic products of graphs. In particular, we give an exact formula in the case of lexicographic product of any two graphs. The NP-hardness of the problem of computing this parameter is derived from the mentioned formula. Graphs GG for which po(G)p_{o}(G) equals the clique number of N(G)\mathcal{N}(G) are also investigated

    On the tractability of some natural packing, covering and partitioning problems

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    In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G=(V,E)G=(V,E) and two "object types" A\mathrm{A} and B\mathrm{B} chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type A\mathrm{A} and one of type B\mathrm{B} in the edge set EE of GG, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition EE into an object of type A\mathrm{A} and one of type B\mathrm{B}? \textbf{Covering problem:} can we cover EE with an object of type A\mathrm{A}, and an object of type B\mathrm{B}? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an ss-tt path PP and an ss'-tt' path PP' that are edge-disjoint. However, many others were not, for example can we find an ss-tt path PEP\subseteq E and a spanning tree TET\subseteq E that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

    Some NP-complete edge packing and partitioning problems in planar graphs

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    Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bern\'ath and Kir\'aly proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be partitioned into a spanning tree and a (not-necessarily spanning) tree. We prove that all three problems remain NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure

    Defensive alliances in graphs: a survey

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    A set SS of vertices of a graph GG is a defensive kk-alliance in GG if every vertex of SS has at least kk more neighbors inside of SS than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive kk-alliance number, Boundary defensive kk-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive kk-alliances, and Defensive kk-alliance free sets.Comment: 25 page

    Partitioning random graphs into monochromatic components

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    Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every rr-colored complete graph can be partitioned into at most r1r-1 monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into rr monochromatic components is possible for sufficiently large rr-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if p(27lognn)1/3p\ge \left(\frac{27\log n}{n}\right)^{1/3}, then a.a.s. in every 22-coloring of G(n,p)G(n,p) there exists a partition into two monochromatic components, and for r2r\geq 2 if p(rlognn)1/rp\ll \left(\frac{r\log n}{n}\right)^{1/r}, then a.a.s. there exists an rr-coloring of G(n,p)G(n,p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in rr-colored complete graphs. We show that if p=ω(1)np=\frac{\omega(1)}{n}, then a.a.s. in every rr-coloring of G(n,p)G(n,p) there exists a monochromatic component of order at least (1o(1))nr1(1-o(1))\frac{n}{r-1}.Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics Volume 24, Issue 1 (2017) Paper #P1.1
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