Abstract
Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G, H) is a mapping f: V(G) → {1, 2, ···, k} such that |f(u) − f(v)| ≥ 2 if uv ∈ E(H) and |f(u) − f(v)| ≥ 1 if uv ∈ E(G)E(H). The backbone chromatic number of (G, H) denoted by χ b (G, H) is the smallest integer k such that (G, H) has a backbone-k-coloring. In this paper, we prove that if G is either a connected triangle-free planar graph or a connected graph with mad(G) < 3, then there exists a spanning tree T of G such that χ b (G, T) ≤ 4.
Similar content being viewed by others
References
Bondy, J.A., Murty, U.S.R. Graph Theory. Springer, Berlin, 2008
Broersma, H.J., Fujisawa, J., Yoshimoto, K. Backbone coloring along perfect matchings. https://www.researchgate.net/publication/228724368 Backbone colorings along perfect matchings, or http://purl.utwente.nl/publications/65891
Broersma, H., Fomin, F.V., Golovach, P.A., Woeginger, G.J. Backbone coloring for graphs: tree and path backbone. J. Grpah Theory, 55(2): 137–152 (2007)
Broersma, H.J., Marchal, L., Paulusma, D., Salman, A.N.M. Improved upper bounds for λ-backbone colorings along matchings and stars. SOFSEM 2007: Theory and Practice of Computer Science, 4362: 188–199 (2007)
Broersma, H.J., Marchal, L., Paulusma, D. Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number. Discuss. Math. Graph Theory, 29(1): 143–162 (2009)
Hajo, Broersma, Fador, V. Fomin, Petr, A. Golovach, Gerhard J. Woeginger. Backbone Coloring for Networks. Proceeding of WG 2003, 2003, LNCS2880, 131–142
Montassier, M., Raspaud, A. (d, 1)-total labeling of graphs with a given maximum average degree. J. Graph Theory, 51: 93–109 (2006)
Wang, Weifang, Bu, Yuehua. Andre Raspaud, and Mickael Montassier. On backbone coloring of graphs. J. Comb. Optim, 23(1): 79–93 (2012)
Salman, A.N.M., Broersma, H.J., Fujisawa, J., Marchal, L., Paulusma, D., Yoshimoto, K. λ-Backbone coloring along pairwise disjoint stars and matchings. Discrete Math., 309(18): 5596–5609 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research supported partially by the National Natural Science Foundation of China (11271334).
Rights and permissions
About this article
Cite this article
Bu, Yh., Zhang, Sm. Backbone coloring for triangle-free planar graphs. Acta Math. Appl. Sin. Engl. Ser. 33, 819–824 (2017). https://doi.org/10.1007/s10255-017-0700-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-017-0700-3