When the vertex coloring of a graph is an edge coloring of its line graph - a rare coincidence

Bujtás, Csilla and Sampathkumar, E. and Tuza, Zsolt and Dominic, C. and Pushpalatha, L (2016) When the vertex coloring of a graph is an edge coloring of its line graph - a rare coincidence. ARS COMBINATORIA, 128. pp. 165-173. ISSN 0381-7032

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Abstract

The 3-consecutive vertex coloring number psi(3c)(G) of a graph G is the maximum number of colors permitted in a coloring of the vertices of G such that the middle vertex of any path P-3 subset of G has the same color as one of the ends of that P-3. This coloring constraint exactly means that no P-3 subgraph of G is properly colored in the classical sense. The 3-consecutive edge coloring number psi(3c)'(G) is the maximum number of colors permitted in a coloring of the edges of G such that the middle edge of any sequence of three edges (in a path P-4 or cycle C-3) has the same color as one of the other two edges. For graphs G of minimum degree at least 2, denoting by L(G) the line graph of G, we prove that there is a bijection between the 3-consecutive vertex colorings of G and the 3-consecutive edge colorings of L(G), which keeps the number of colors unchanged, too. This implies that psi(3c)(G) = psi(3c)'(L(G)); i.e., the situation is just the opposite of what one would expect for first sight.

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