\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Adjacent vertex distinguishing edge-colorings and total-colorings of the Cartesian product of graphs

Abstract Related Papers Cited by
  • Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ □ $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.
    Mathematics Subject Classification: Primary: 05C35, 05C75; Secondary: 05C60.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    S. Akbari, H. Bidkhori and N. Nosrati, r-Strong edge colorings of graphs, Discrete Math., 306 (2006), 3005-3010.doi: 10.1016/j.disc.2004.12.027.

    [2]

    P. N. Balister, E. Győri, J. Lehel and R. H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math., 21 (2007), 237-250.doi: 10.1137/S0895480102414107.

    [3]

    J-L. Baril, H. Kheddouci and O. Togni, Adjacent vertex distinguishing edge-colorings of meshes, Australasian Journal of Combinatorics, 35 (2006), 89-102.

    [4]

    J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976.

    [5]

    M. Chen and X. Guo, Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes, Information Processing Letters, 109 (2009), 599-602.doi: 10.1016/j.ipl.2009.02.006.

    [6]

    K. Edwards, M. Horňák and M. Woźniak, On the neighbor-distinguishing index of a graph, Graphs Combin., 22 (2006), 341-350.doi: 10.1007/s00373-006-0671-2.

    [7]

    H. Hatami, ∆+300 is a bound on the adjacent vertex distinguishing edge chromatic number, J. Combin. Theory Ser. B, 95 (2005), 246-256.doi: 10.1016/j.jctb.2005.04.002.

    [8]

    S. Tian and P. Chen, On adjacent vertex-distinguishing total coloring of two classes of product graphs, Journal of Mathematical Research and Exposition, 27 (2007), 733-737.

    [9]

    H. Wang, On the adjacent vertex-distinguishing total chromatic numbers of the graphs with ∆(G)=3, Journal of Combinatorial Optimization, 14 (2007), 87-109.doi: 10.1007/s10878-006-9038-0.

    [10]

    H. P. Yap, Total Coloring of Graph, Springer Verlag, New York, 1996.

    [11]

    Z. Zhang, X. Chen, J. Li, B. Yao, X. Lu and J. Wang, On adjacent-vertex-distinguishing total coloring of graphs, Science in China Series A, Mathematics, 48 (2005), 289-299.doi: 10.1360/03YS0207.

    [12]

    Z. Zhang, L. Liu and J. Wang, Adjacent strong edge coloring of graphs, Applied Mathematics Letters, 15 (2002), 623-626.doi: 10.1016/S0893-9659(02)80015-5.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(106) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return