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Modular edge colorings of Mycielskian graphs | ||
| Transactions on Combinatorics | ||
| مقاله 35، دوره 4، شماره 3، آذر 2015، صفحه 53-61 اصل مقاله (213.97 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2015.7309 | ||
| نویسندگان | ||
| N. Paramaguru؛ R. Sampathkumar* | ||
| Annamalai University | ||
| چکیده | ||
| Let $G$ be a connected graph of order $3$ or more and $c:E(G)\rightarrow\mathbb{Z}_k$ ($k\ge 2$) a $k$-edge coloring of $G$ where adjacent edges may be colored the same. The color sum $s(v)$ of a vertex $v$ of $G$ is the sum in $\mathbb{Z}_k$ of the colors of the edges incident with $v.$ The $k$-edge coloring $c$ is a modular $k$-edge coloring of $G$ if $s(u)\ne s(v)$ in $\mathbb{Z}_k$ for all pairs $u,$ $v$ of adjacent vertices of $G.$ The modular chromatic index $\chi'_m(G)$ of $G$ is the minimum $k$ for which $G$ has a modular $k$-edge coloring. The Mycielskian of $G\,=\,(V,E)$ is the graph $\mathscr{M}(G)$ with vertex set $V\cup V'\cup\{u\},$ where $V'=\{v':v\in V\},$ and edge set $E\cup\{xy':xy\in E\}\cup\{v'u:v'\in V'\}.$ It is shown that $\chi'_m(\mathscr{M}(G))\,=\,\chi(\mathscr{M}(G))$ for some bipartite graphs, cycles and complete graphs. | ||
| کلیدواژهها | ||
| modular edge coloring؛ modular chromatic index؛ Mycielskian of a graph | ||
| مراجع | ||
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R. Balakrishnan and K. Ranganathan (2012) A textbook of graph theory Second edition, Universitext, Springer, New York
R. Jones, K. Kolasinski, F. Okamoto and P. Zhang (2011) Modular Neighbor-Distinguishing Edge Colorings of Graphs J. Combin. Math. Combin. Comput. 76, 159-175
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