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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 from the direct product $G times P_{n,m}$ by identifying $V(G) times {n}$ into a single vertex $(star, n)$, identifying $V(G) times {-m}$ into a single vertex $(star, -m)$, and adding an edge connecting $(star, -m)$ and $(star, n)$. This paper determines the fractional chromatic number of $Delta_{n,m}(G)$. In particular, if $n < m$ or $n=m$ is even, then $chi_f(Delta_{n,m}(G)) = chi_f(Delta_n(G))$, where $Delta_n(G)$ is the $n$th cone over $G$. If $n=m$ is odd, then $chi_f(Delta_{n,m}(G)) > chi_f(Delta_n(G))$. The chromatic number of $Delta_{n,m}(G)$ is also discussed.
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 $
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 )