### 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 |
---|---|

Pages (from-to) | 165-173 |

Number of pages | 9 |

Journal | Ars Combinatoria |

Volume | 128 |

Publication status | Published - Jul 1 2016 |

### Fingerprint

### Keywords

- 3-consecutive edge coloring
- 3-consecutive vertex coloring
- Line graph
- Matching
- Stable k-separator

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Ars Combinatoria*,

*128*, 165-173.

**When the vertex coloring of a graph is an edge coloring of its line graph - A rare coincidence.** / Bujtás, Csilla; Sampathkumar, E.; Tuza, Z.; Dominic, Charles; Pushpalatha, L.

Research output: Contribution to journal › Article

*Ars Combinatoria*, vol. 128, pp. 165-173.

}

TY - JOUR

T1 - When the vertex coloring of a graph is an edge coloring of its line graph - A rare coincidence

AU - Bujtás, Csilla

AU - Sampathkumar, E.

AU - Tuza, Z.

AU - Dominic, Charles

AU - Pushpalatha, L.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

KW - 3-consecutive edge coloring

KW - 3-consecutive vertex coloring

KW - Line graph

KW - Matching

KW - Stable k-separator

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UR - http://www.scopus.com/inward/citedby.url?scp=85031325340&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85031325340

VL - 128

SP - 165

EP - 173

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -