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L-Factors and Adjacent Vertex-Distinguishing Edge-Weighting

Published online by Cambridge University Press:  28 May 2015

Yinghua Duan ,
Hongliang Lu  and
Qinglin Yu
Yinghua Duan*
Affiliation:
Department of Mathematics, Beijing National Day School, Beijing, PR China
Hongliang Lu*
Affiliation:
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China
Qinglin Yu*
Affiliation:
Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, BC, Canada
*
Corresponding author. Email: yinghua115@mail.nankai.edu.cn
Corresponding author. Email: luhongliang215@sina.com
Corresponding author. Email: yu@tru.ca
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Abstract

An edge-weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on an edge-weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that the edge-weighting problem has a close relationship with special factors of the graphs. In this paper, we generalise several earlier results on the existence of factors with pre-specified degrees and hence investigate the edge-weighting problem — and in particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weighting.

Keywords

05C7005C15Edge-weightingvertex-coloringL-factor
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 2 , May 2012 , pp. 83 - 93
Copyright
Copyright © Global-Science Press 2012

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References

[1]Addario-Berry, L., Aldred, R. E. L., Dalal, K. and Reed, B. A., Vertex coloring edge partitions, J. Combinatorial Theory Ser. B, 94 (2005), 237–244.CrossRefGoogle Scholar
[2]Addario-Berry, L., Dalal, K., McDiarmid, C., Reed, B. A. and Thomason, A., Vertex-coloring edge-weightings, Combintorica, 27 (2007), 1–12.Google Scholar
[3]Addario-Berry, L., Dalal, K. and Reed, B. A., Degree constrained subgraphs, Discrete Applied Math., 156 (2008), 1168–1174.CrossRefGoogle Scholar
[4]Aigner, M., Triesch, E. and Tuza, Zs., Irregular assignments and vertex-distinguishing edge-colorings of graphs, Ann. Discrete Math., 52, North-Holland, Amsterdam, 1992, 1–9.Google Scholar
[5]Balister, P. N., Riordan, O. M. and Schelp, R. H., Vertex-distinguishing edge colorings of graphs, J. Graph Theory, 42 (2003), 95–109.Google Scholar
[6]Bollobás, B., Modern Graph Theory, 2nd Edition, Springer-Verlag New York, Inc. 1998.Google Scholar
[7]Burris, A. C. and Schelp, R. H., Vertex-distinguishing proper edge colorings, J. Graph Theory, 26 (1997), 73–82.Google Scholar
[8]Chang, G. J., Lu, C., Wu, J. and Yu, Q. L., Vertex-coloring edge- weighting of graphs, Taiwanese Journal of Mathematics, 15 (2011), 1807–1813Google Scholar
[9]Lu, H. L., Yu, Q. L. and Zhang, C. Q., Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22–27.Google Scholar
[10]Edwards, K., The harmonioous chromatic number of bounded degree graphs, J. London Math. Soc., 55 (1997), 435–447.Google Scholar
[11]Heinrich, K., Hell, P., Kirkpatrick, D. G. and Liu, G. Z., A simple existence criterion for (g < f)-factors, Discrete Math., 85 (1990), 313–317.Google Scholar
[12]Kalkowski, M., Karoński, M., and Pfender, F., Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture, J. Combinatorial Theory Ser. B, 100 (2010), 347–349.Google Scholar
[13]Karoński, M., Łuczak, T. and Thomason, A., Edge weights and vertex colors, J. Combinatorial Theory Ser. B, 91 (2004), 151–157.Google Scholar
[14]Lovász, L., The factorization of graphs (II), Acta Math. Hungar., 23 (1972), 223–246.CrossRefGoogle Scholar
[15]Wang, T. and Yu, Q. L., A note on vertex-coloring 13-edge-weighting, Frontier Math. in China, 3 (2008), 1–7.Google Scholar