11,760 research outputs found

    On r-Dynamic Chromatic Number of the Corronation of Path and Several Graphs

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    This study is a natural extension of k-proper coloring of any simple and connected graph G. By an r-dynamic coloring of a graph G, we mean a proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. The r-dynamic chromatic number, written as r(G), is the minimum k such that graph G has an r-dynamic k-coloring. In this paper we will study the r-dynamic chromatic number of the coronation of path and several graph. We denote the corona product of G and H by G⨀▒H. We will obtain the r-dynamic chromatic number of χ_r (P_n⨀P_m ),χ_r (P_n⨀C_m )"and " χ_r (P_n⨀W_m ) for m, n>= 3

    On the oriented chromatic number of dense graphs

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    Let GG be a graph with nn vertices, mm edges, average degree δ\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 δlogn\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 δ(2+ϵ)logn\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 (clognc\log n)-regular for some constant c>2c>2, in which case the oriented chromatic number is between Ω(nlogn)\Omega(\sqrt{n\log n}) and O(nlogn)\mathcal{O}(\sqrt{n}\log n)

    Game chromatic number of Cartesian and corona product graphs

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    The game chromatic number χg\chi_g is investigated for Cartesian product GHG\square H and corona product GHG\circ H of two graphs GG and HH. The exact values for the game chromatic number of Cartesian product graph of S3SnS_{3}\square S_{n} is found, where SnS_n is a star graph of order n+1n+1. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let PmP_m be the path graph on mm vertices and CnC_n be the cycle graph on nn vertices. We have determined the exact values for the game chromatic number of corona product graphs PmK1P_{m}\circ K_{1} and PmCnP_{m}\circ C_{n}
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