Abstract
A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we give some upper bounds on linear chromatic number for plane graphs with respect to their girth, that improve some results of Raspaud and Wang (2009).
Similar content being viewed by others
References
Alon N, McDiarmid C J H, Reed B A. Acyclic coloring of graphs. Rand Struct Algor, 1991, 2: 277–288
Alon N, Zaks A. Algorithmic aspects of acyclic edge colorings. Algorithmica, 2002, 32: 611–614
Borodin O V. On acyclic colorings of planar graphs. Discrete Math, 1979, 25: 211–236
Borodin O V, Fon-Der Flaass D G, Kostochka A V, et al. Acyclic list 7-coloring of planar graphs. J Graph Theory, 2002, 40: 83–90
Esperet L, Montassier M, Raspaud A. Linear choosability of graphs. Discrete Math, 2008, 308: 3938–3950
Gerke S, Raemy M. Generalised acyclic edge colourings of graphs with large girth. Discrete Math, 2007, 307: 1668–1671
Grünbaum B. Acyclic colorings of planar graphs. Israel J Math, 1973, 14: 390–408
Hind H, Molloy M, Reed B. Colouring a graph frugally. Combinatorica, 1997, 17: 469–482
Hou J F, Wu J L, Liu G Z, et al. Acyclic edge colorings of planar graphs and series parallel graphs. Sci China Ser A, 2009, 52: 605–616
Kostochka A V. Acyclic 6-coloring of planar graphs. Metody Diskret Anal, 1976, 28: 40–56
Manu B, Sunil C L. Acyclic edge coloring of subcubic graphs. Discrete Math, 2008, 308: 6650–6653
Mitchem J. Every planar graph has an acyclic 8-coloring. Duke Math J, 1974, 14: 177–181
Muthu R, Narayanan N, Subramanian C R. Acyclic edge colouring of outerplanar graphs. In: Lecture Notes in Computer Science, vol. 4508. Algorithm Aspects in Information and Management. Berlin-Heidelberg: Springer-Verlag, 2009, 144–152
Raspaud A, Wang W F. Linear coloring of planar graphs with large girth. Discrete Math, 2009, 309: 5678–5686
Wang W F, Li C. Linear coloring of graphs embeddable in a surface of nonnegative characteristic. Sci China Ser A, 2009, 52: 991–1003
Yuster R. Linear coloring of graphs. Discrete Math, 1998, 185: 293–297
Zhang Z F, Chen X E, Li J W, et al. On adjacent-vertex-distinguishing total coloring of graphs. Sci China Ser A, 2005, 48: 289–299
Zhang Z F, Cheng H, Yao B, et al. On the adjacent-vertex-strongly-distinguishing total coloring of graphs. Sci China Ser A, 2008, 51: 427–436
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dong, W., Xu, B. & Zhang, X. Improved bounds on linear coloring of plane graphs. Sci. China Math. 53, 1895–1902 (2010). https://doi.org/10.1007/s11425-010-3073-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-3073-0