ArticleOriginal scientific text
Title
Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six
Authors 1, 2, 1
Affiliations
- Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China
- Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan
Abstract
An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring o G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ'ₐ(G). We prove that χ'ₐ(G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges whose girth is at least 6. This gives new evidence to a conjecture proposed in [Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett., 15 (2002) 623-626.]
Keywords
edge-coloring, vertex-distinguishing, planar graph
Bibliography
- M. Aigner, E. Triesch and Z. Tuza, Irregular assignments and vertex-distinguishing edge-colorings of graphs, in: Proceedings of Combinatorics '90, A. Barlotti et al., eds. (North-Holland, Amsterdam, 1992) 1-9.
- S. Akbari, H. Bidkhori and N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006) 3005-3010, doi: 10.1016/j.disc.2004.12.027.
- P.N. Balister, E. Gyori, J. Lehel and R. H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math. 21 (2007) 237-250, doi: 10.1137/S0895480102414107.
- P.N. Balister, O.M. Riordan and R.H. Schelp, Vertex-distinguishing edge colorings of graphs, J. Graph Theory 42 (2003) 95-109, doi: 10.1002/jgt.10076.
- J.-L. Baril, H. Kheddouci and O. Togni, Adjacent vertex distinguishing edge-colorings of meshes, Australas. J. Combin. 35 (2006) 89-102.
- J.-L. Baril and O. Togni, Neighbor-distinguishing k-tuple edge-colorings of graphs, Discrete Math. 309 (2009) 5147-5157, doi: 10.1016/j.disc.2009.04.003.
- A.C. Burris and R.H. Schelp, Vertex-distinguishing proper edge-colorings, J. Graph Theory 26 (1997) 70-82, doi: 10.1002/(SICI)1097-0118(199710)26:2<73::AID-JGT2>3.0.CO;2-C
- K. Edwards, M. Hornák and M. Woźniak, On the neighbour-distinguishing index of a graph, Graphs Combin. 22 (2006) 341-350, doi: 10.1007/s00373-006-0671-2.
- O. Favaron, H. Li and R.H. Schelp, Strong edge colorings of graphs, Discrete Math. 159 (1996) 103-109, doi: 10.1016/0012-365X(95)00102-3.
- H. Hatami, Δ+300 is a bound on the adjacent vertex distinguishing edge chromatic number, J. Combin. Theory (B) 95 (2005) 246-256, doi: 10.1016/j.jctb.2005.04.002.
- V.G. Vizing, On an estimate of the chromatic index of a p-graph, Diskret Analiz. 3 (1964) 25-30.
- Z. Zhang, L. Liu and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626, doi: 10.1016/S0893-9659(02)80015-5.
Pages:
429-439
Main language of publication
English
Received
2009-11-18
Accepted
2010-03-27
Published
2011
DOI: 10.7151/dmgt.1556
Exact and natural sciences