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For a simple graph $G$, let $chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $Delta geq 4$, if $G$ has maximum degree at most $Delta$ and is $K_{Delta}$-free, then $chi_f(G) leq Delta-tfrac{1}{8}$ unless $G= C^2_8$ or $G = C_5boxtimes K_2$. This im proves the result in [King, Lu, and Peng, SIAM J. Discrete Math., 26(2) (2012), pp. 452-471] for $Delta geq 4$ and the result in [Katherine and King, SIAM J.Discrete Math., 27(2) (2013), pp. 1184-1208] for $Delta in {6,7,8}$.
Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices ${-m, -(m-1), ldots, 0, ldots, n}$ by adding a loop at vertex $ 0$. The double cone $Delta_{n,m}(G)$ over a graph $G$ is obtained
By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]={1,2,ldots, n}$ for some integers $nge kge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in ad
A signed graph is a pair $(G, sigma)$, where $G$ is a graph and $sigma: E(G) to {+, -}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular colorin
Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.
Let $G$ be a simple graph with maximum degree $Delta(G)$ and chromatic index $chi(G)$. A classic result of Vizing indicates that either $chi(G )=Delta(G)$ or $chi(G )=Delta(G)+1$. The graph $G$ is called $Delta$-critical if $G$ is connected, $chi(G )