Opuscula Math. 36, no. 5 (2016), 603-612
http://dx.doi.org/10.7494/OpMath.2016.36.5.603

Opuscula Mathematica

# M2-edge colorings of dense graphs

Jaroslav Ivančo

Abstract. An edge coloring $$\varphi$$ of a graph $$G$$ is called an $$\mathrm{M}_i$$-edge coloring if $$|\varphi(v)|\leq i$$ for every vertex $$v$$ of $$G$$, where $$\varphi(v)$$ is the set of colors of edges incident with $$v$$. Let $$\mathcal{K}_i(G)$$ denote the maximum number of colors used in an $$\mathrm{M}_i$$-edge coloring of $$G$$. In this paper we establish some bounds of $$\mathcal{K}_2(G)$$, present some graphs achieving the bounds and determine exact values of $$\mathcal{K}_2(G)$$ for dense graphs.

Keywords: edge coloring, dominating set, dense graphs.

Mathematics Subject Classification: 05C15.

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