Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T17:51:41.992Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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