Abstract
A graph G=(V, E) is (k, k′)-total weight choosable if the following is true: For any (k, k′)-total list assignment L that assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights, there is a proper L-total weighting, i.e., a mapping f:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ e∈E(u) f(e)+f(u)≠∑ e∈E(v) f(e)+f(v). This Paper introduces a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of the form K n,m,1,1,...,1 are (2,2)-total weight choosable and complete bipartite graphs other than K 2 are (1,2)-total weight choosable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Addario-Berry, R. E. L. Aldred, K. Dalai and B. A. Reed, Vertex colouring edge partitions, J. Combin. Theory Ser. B, 94 (2005), 237–244.
L. Addario-Berry, K. Dalai and B. A. Reed, Degree constrianed subgraphs, Proceedings of GRACO2005, Volume 19, Electron. Notes Discrete Math., Amsterdam (2005), 257–263, Elsevier.
L. Addario-Berry, K. Dalai, C. McDiarmid, B. A. Reed and A. Thomason, Vertex-colouring edge-weightings, Combinatorica, 27 (2007), 1–12.
N. Alon and M. Tarsi, A nowhere zero point in linear mappings, Combinatorica, 9 (1989), 393–395.
N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica, 12 (1992), 125–134.
N. Alon and M. Tarsi, Combinatorial Nullstellensatz, Combin. Prob. Comput., 8 (1999), 7–29.
T. Bartnicki, J. Grytczuk and S. Niwczyk, Weight choosability of graphs, Journal of Graph Theory, 60(3) (2009), 242–256.
G. Chang, C. Lu, J. Wu and Q. Yu, Vertex coloring 2-edge weighting of bipartite graphs, preprint, 2007.
M. Karonski, T. Luczak and A. Thomason, Edge weights and vertex colour, J. Combin. Theory Ser. B, 91 (2004), 151–157.
M. Kalkowski, M. Karońs ki and F. Pfender, Vertex-coloring edge-weightings: towards the 1-2-3-Conjecture, Journal of Combinatorial Theory, Ser. B, to appear.
J. Przybylo and M. Wozniak, 1,2-conjecture, Preprint MD 024, www.ii.uj.edu.pl/preMD/
J. Przybylo and M. Wozniak, 1,2-conjecture II, Preprint MD 026, www.ii.uj.edu.pl/preMD/
T. Wong and X. Zhu, Total weight choosability of graphs, Journal of Graph Theory, to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Wong, TL., Zhu, X., Yang, D. (2010). List Total Weighting of Graphs. In: Katona, G.O.H., Schrijver, A., Szőnyi, T., Sági, G. (eds) Fete of Combinatorics and Computer Science. Bolyai Society Mathematical Studies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13580-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-13580-4_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13579-8
Online ISBN: 978-3-642-13580-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)