The Circular Chromatic Number of the Mycielski's Graph M
نویسندگان
چکیده
As a natural generalization of chromatic number of a graph, the circular chromatic number of graphs (or the star chromatic number) was introduced by A.Vince in 1988. Let M (G) denote the tth iterated Mycielski graph of G. It was conjectured by Chang, Huang and Zhu(Discrete mathematics,205(1999), 23-37) that for all n ≥ t+2, χc(M (Kn)) = χ(M (Kn)) = n+ t. In 2004, D.D.F. Liu proved the conjecture when t ≥ 2, n ≥ 2+2t−2. In this paper,we show that the result can be strengthened to the following: if t ≥ 4, n ≥ 11 122 +2t+ 1 3 , then χc(M (Kn)) = χ(M (Kn)). Keyword: circular chromatic number, complete graph, Mycielski graph
منابع مشابه
Computing Multiplicative Zagreb Indices with Respect to Chromatic and Clique Numbers
The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...
متن کاملChromatic polynomials of some nanostars
Let G be a simple graph and (G,) denotes the number of proper vertex colourings of G with at most colours, which is for a fixed graph G , a polynomial in , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.
متن کاملThe fractional chromatic number of mycielski's graphs
The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence G n of triangle-free graphs with (G n ) = n. In this note, we calculate the fractional chromatic number of G n and show that this sequence of numbers satis es the unexpected recurrence a n+1 = a n + 1 a n .
متن کاملThe distinguishing chromatic number of bipartite graphs of girth at least six
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...
متن کاملThe locating-chromatic number for Halin graphs
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...
متن کامل