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 P3 C G has the same color as one of the ends of that P3. This coloring constraint exactly means that no P3 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 P4 or cycle C3) 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.
|Number of pages||9|
|Publication status||Published - júl. 1 2016|
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