A note on the star chromatic number
Journal of Graph Theory
Star chromatic numbers and products of graphs
Journal of Graph Theory
The star chromatic number of a graph
Journal of Graph Theory
Some theorems concerning the star chromatic number of a graph
Journal of Combinatorial Theory Series B
Star chromatic numbers of some planar graphs
Journal of Graph Theory
Circular chromatic numbers of Mycielski's graphs
Discrete Mathematics
Circular chromatic number: a survey
Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
The circular chromatic number of the Mycielskian of G dk
Journal of Graph Theory
Several parameters of generalized Mycielskians
Discrete Applied Mathematics
Colorful subgraphs in Kneser-like graphs
European Journal of Combinatorics
Several parameters of generalized Mycielskians
Discrete Applied Mathematics
Multiple Coloring of Cone Graphs
SIAM Journal on Discrete Mathematics
Hi-index | 0.00 |
In this paper, we introduce a graph transformation analogous to that of Mycielski. Given a graph G and any integer m, one can transform G into a new graph µm(G), the generalized Mycielskian of G. Many basic properties of µm(G) were established in (Lam et al., Some properties of generalized Mycielski's graphs, to appear). Here we completely determine the circular chromatic number of µm(Kn) for any m(≥0) and n(≥ 2). We prove that for any odd integer n ≥ 3 and any nonnegative integer m, χc (µm(Kn)) = χ(µm(Kn)) = n + 1. This answers part of the question raised by Zhou (J. Combin. Theory Ser. B 70 (1997) 245) or that by Zhu (Discrete Math. 229 (2001) 371). Because µm(K3), for arbitrary m, is a planar graph with connectivity 3 and maximum degree 4, it provides another counterexample to a question asked by Vince (J. Graph Theory 17 (1993) 349). For any positive even number n(≥2) and any nonnegative integer m, we show that χc(µm(Kn)) = n + (1/t), where t = ⌊2m/n⌋ + 1. This gives a family of arbitrarily large critical graphs G with high connectivity and small maximum degree for which χc(G) can be arbitrarily close to χ(G) - 1.