Abstract
Let G=(V(G),E(G)) be a simple graph. Given non-negative integers r,s, and t, an [r,s,t]-coloring of G is a mapping c from V(G)∪E(G) to the color set {0,1,…,k−1} such that |c(v i )−c(v j )|≥r for every two adjacent vertices v i ,v j , |c(e i )−c(e j )|≥s for every two adjacent edges e i ,e j , and |c(v i )−c(e j )|≥t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number χ r,s,t (G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. We determine χ r,s,t (K n,n ) in all cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley-Interscience, New York (1995)
Kemnitz, A., Marangio, M.: [r,s,t]-Coloring of graphs. Discrete Math. 307, 199–207 (2007)
Yap, H.P.: Total Colorings of Graphs. Lecture Notes in Mathematics, vol. 1623. Springer, Berlin (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by HENSF(A2007000002) and NNSF(10871058) of China.
Rights and permissions
About this article
Cite this article
Xu, C., Ma, X. & Hua, S. [r,s,t]-Coloring of K n,n . J. Appl. Math. Comput. 31, 45–50 (2009). https://doi.org/10.1007/s12190-008-0190-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-008-0190-9