# [r, s, t; f]-COLORING OF GRAPHS

• Yu, Yong (SCHOOL OF MATHEMATICS SHANDONG UNIVERSITY) ;
• Liu, Guizhen (SCHOOL OF MATHEMATICS SHANDONG UNIVERSITY)
• Accepted : 2009.10.13
• Published : 2011.01.01
• 57 10

#### Abstract

Let f be a function which assigns a positive integer f(v) to each vertex v $\in$ V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v $\in$ V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by ${\chi}'_f$(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) $\bigcup$ E(G) to the color set C = {0, 1, $\ldots$; k - 1} such that |c($v_i$) - c($v_j$ )| $\geq$ r for every two adjacent vertices $v_i$ and $v_j$, |c($e_i$ - c($e_j$)| $\geq$ s and ${\alpha}(v_i)$ $\leq$ f($v_i$) for all $v_i$ $\in$ V (G), ${\alpha}$ $\in$ C where ${\alpha}(v_i)$ denotes the number of ${\alpha}$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, |c($e_i$)-c($v_j$)| $\geq$ t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by ${\chi}_{r,s,t;f}$ (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.

#### Keywords

f-coloring;[r,s,t]-coloring;[r,s,t;f]-coloring;f-total coloring;[r,s,t;f]-chromatic number

#### Acknowledgement

Supported by : NSFC, NNSF, RSDP

#### References

1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
2. R. L. Brooks, On colouring the nodes of a network, Proc. Cambridge Philos. Soc. 37 (1941), 194-197. https://doi.org/10.1017/S030500410002168X
3. L. Dekar, et al., [r, s, t]-coloring of trees and bipartite graphs, Discrete Math. (2008) doi:10.1016/j.disc.2008.09.021. https://doi.org/10.1016/j.disc.2008.09.021
4. S. L. Hakimi and O. Kariv, A generalization of edge-coloring in graphs, J. Graph Theory 10 (1986), no. 2, 139-154. https://doi.org/10.1002/jgt.3190100202
5. F. Havet and M. L. Yu, (p, 1)-total labelling of graphs, Discrete Math. 308 (2008), no. 4, 496-513. https://doi.org/10.1016/j.disc.2007.03.034
6. A. Kemnitz and M. Marangio, [r, s, t]-colorings of graphs, Discrete Math. 307 (2007), no. 2, 199-207. https://doi.org/10.1016/j.disc.2006.06.030
7. X. Zhang and G. Liu, The classification of complete graphs $K_n$ on f-coloring, J. Appl. Math. Comput. 19 (2005), no. 1-2, 127-133. https://doi.org/10.1007/BF02935792
8. X. Zhang and G. Liu, Some sufficient conditions for a graph to be of $C_f$ 1, Appl. Math. Lett. 19 (2006), no. 1, 38-44. https://doi.org/10.1016/j.aml.2005.03.006
9. X. Zhang and G. Liu, Some graphs of class 1 for f-colorings, Appl. Math. Lett. 21 (2008), no. 1, 23-29. https://doi.org/10.1016/j.aml.2007.02.009