### 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} C 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.

Original language | English |
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Pages (from-to) | 165-173 |

Number of pages | 9 |

Journal | Ars Combinatoria |

Volume | 128 |

Publication status | Published - júl. 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Ars Combinatoria*,

*128*, 165-173.