39 research outputs found

    Some Results on incidence coloring, star arboricity and domination number

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    Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.Comment: 8 page

    The Incidence Chromatic Number of Toroidal Grids

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    An incidence in a graph GG is a pair (v,e)(v,e) with vV(G)v \in V(G) and eE(G)e \in E(G), such that vv and ee are incident. Two incidences (v,e)(v,e) and (w,f)(w,f) are adjacent if v=wv=w, or e=fe=f, or the edge vwvw equals ee or ff. The incidence chromatic number of GG is the smallest kk for which there exists a mapping from the set of incidences of GG to a set of kk colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n=CmCnT_{m,n}=C_m\Box C_n equals 5 when m,n0(mod5)m,n \equiv 0 \pmod 5 and 6 otherwise.Comment: 16 page

    Graph Treewidth and Geometric Thickness Parameters

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    Consider a drawing of a graph GG in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of GG, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth kk, the maximum thickness and the maximum geometric thickness both equal k/2\lceil{k/2}\rceil. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth kk, the maximum book thickness equals kk if k2k \leq 2 and equals k+1k+1 if k3k \geq 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

    On globally sparse Ramsey graphs

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    We say that a graph GG has the Ramsey property w.r.t.\ some graph FF and some integer r2r\geq 2, or GG is (F,r)(F,r)-Ramsey for short, if any rr-coloring of the edges of GG contains a monochromatic copy of FF. R{\"o}dl and Ruci{\'n}ski asked how globally sparse (F,r)(F,r)-Ramsey graphs GG can possibly be, where the density of GG is measured by the subgraph HGH\subseteq G with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs FF. In this work we determine the Ramsey density up to some small error terms for several cases when FF is a complete bipartite graph, a cycle or a path, and r2r\geq 2 colors are available

    Planar Ramsey graphs

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    We say that a graph HH is planar unavoidable if there is a planar graph GG such that any red/blue coloring of the edges of GG contains a monochromatic copy of HH, otherwise we say that HH is planar avoidable. I.e., HH is planar unavoidable if there is a Ramsey graph for HH that is planar. It follows from the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar unavoidable then it is bipartite and outerplanar. We prove that the cycle on 44 vertices and any path are planar unavoidable. In addition, we prove that all trees of radius at most 22 are planar unavoidable and there are trees of radius 33 that are planar avoidable. We also address the planar unavoidable notion in more than two colors

    The acircuitic directed star arboricity of subcubic graphs is at most four

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    AbstractA directed star forest is a forest all of whose components are stars with arcs emanating from the center to the leaves. The acircuitic directed star arboricity of an oriented graph G (that is a digraph with no opposite arcs) is the minimum number of arc-disjoint directed star forests whose union covers all arcs of G and such that the union of any two such forests is acircuitic. We show that every subcubic graph has acircuitic directed star arboricity at most four
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