5,265 research outputs found

    On star edge colorings of bipartite and subcubic graphs

    Full text link
    A star edge coloring of a graph is a proper edge coloring with no 22-colored path or cycle of length four. The star chromatic index χst(G)\chi'_{st}(G) of GG is the minimum number tt for which GG has a star edge coloring with tt colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most 33. We also consider bipartite graphs GG where all vertices in one part have maximum degree 22 and all vertices in the other part has maximum degree bb. Let kk be an integer (k1k\geq 1), we prove that if b=2k+1b=2k+1 then χst(G)3k+2\chi'_{st}(G) \leq 3k+2; and if b=2kb=2k, then χst(G)3k\chi'_{st}(G) \leq 3k; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 66; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page

    Vertex-Coloring with Star-Defects

    Full text link
    Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors

    Star 5-edge-colorings of subcubic multigraphs

    Full text link
    The star chromatic index of a multigraph GG, denoted χs(G)\chi'_{s}(G), is the minimum number of colors needed to properly color the edges of GG such that no path or cycle of length four is bi-colored. A multigraph GG is star kk-edge-colorable if χs(G)k\chi'_{s}(G)\le k. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star 77-edge-colorable, and conjectured that every subcubic multigraph should be star 66-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than 7/37/3 is star list-55-edge-colorable. It is known that a graph with maximum average degree 14/514/5 is not necessarily star 55-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than 12/512/5 is star 55-edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap with arXiv:1701.0410

    Ramsey-nice families of graphs

    Get PDF
    For a finite family F\mathcal{F} of fixed graphs let Rk(F)R_k(\mathcal{F}) be the smallest integer nn for which every kk-coloring of the edges of the complete graph KnK_n yields a monochromatic copy of some FFF\in\mathcal{F}. We say that F\mathcal{F} is kk-nice if for every graph GG with χ(G)=Rk(F)\chi(G)=R_k(\mathcal{F}) and for every kk-coloring of E(G)E(G) there exists a monochromatic copy of some FFF\in\mathcal{F}. It is easy to see that if F\mathcal{F} contains no forest, then it is not kk-nice for any kk. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F\mathcal{F} that contains at least one forest, and for all kk0(F)k\geq k_0(\mathcal{F}) (or at least for infinitely many values of kk), F\mathcal{F} is kk-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family F\mathcal{F} containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

    Distance-two coloring of sparse graphs

    Full text link
    Consider a graph G=(V,E)G = (V, E) and, for each vertex vVv \in V, a subset Σ(v)\Sigma(v) of neighbors of vv. A Σ\Sigma-coloring is a coloring of the elements of VV so that vertices appearing together in some Σ(v)\Sigma(v) receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set Σ(v)\Sigma(v), denoted by ρ(Σ)\rho(\Sigma). In this paper we study graph classes FF for which there is a function ff, such that for any graph GFG \in F and any Σ\Sigma, there is a Σ\Sigma-coloring using at most f(ρ(Σ))f(\rho(\Sigma)) colors. It is proved that if such a function exists for a class FF, then ff can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes.Comment: 13 pages - revised versio
    corecore