592 research outputs found

    The harmonious chromatic number of almost all trees

    Get PDF

    The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs

    Full text link
    The edge-distinguishing chromatic number (EDCN) of a graph GG is the minimum positive integer kk such that there exists a vertex coloring c:V(G)→{1,2,
,k}c:V(G)\to\{1,2,\dotsc,k\} whose induced edge labels {c(u),c(v)}\{c(u),c(v)\} are distinct for all edges uvuv. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.Comment: 23 pages, 1 figur

    On the oriented chromatic number of dense graphs

    Get PDF
    Let GG be a graph with nn vertices, mm edges, average degree ÎŽ\delta, and maximum degree Δ\Delta. The \emph{oriented chromatic number} of GG is the maximum, taken over all orientations of GG, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which Ύ≄log⁥n\delta\geq\log n. We prove that every such graph has oriented chromatic number at least Ω(n)\Omega(\sqrt{n}). In the case that Ύ≄(2+Ï”)log⁥n\delta\geq(2+\epsilon)\log n, this lower bound is improved to Ω(m)\Omega(\sqrt{m}). Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when GG is (clog⁥nc\log n)-regular for some constant c>2c>2, in which case the oriented chromatic number is between Ω(nlog⁥n)\Omega(\sqrt{n\log n}) and O(nlog⁥n)\mathcal{O}(\sqrt{n}\log n)

    Connection Matrices and the Definability of Graph Parameters

    Get PDF
    In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky (2008 and 2009), and demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers

    The b-Chromatic Number of Star Graph Families

    Get PDF
    In this paper, we investigate the b-chromatic number of central graph, middle graph and total graph of star graph, denoted by C(K1,n), M(K1,n)  and  T(K1,n) respectively. We discuss the relationship between b-chromatic number with some other types of chromatic numbers such as chromatic number, star chromatic number and equitable chromatic number

    Defective and Clustered Graph Colouring

    Full text link
    Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" dd if each monochromatic component has maximum degree at most dd. A colouring has "clustering" cc if each monochromatic component has at most cc vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding KtK_t as a minor, graphs excluding Ks,tK_{s,t} as a minor, and graphs excluding an arbitrary graph HH as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

    Subject Index Volumes 1–200

    Get PDF
    • 

    corecore