Let ∆(G) be the maximum degree of a graph G. Brooks’ theorem states that the only connected graphs with chromatic number χ(G) = ∆(G) + 1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks’ theorem in this paper. Namely, we classify all connected graphs G such that the fractional chromatic number χf (G) is at least ∆(G). These graphs are complete graphs, odd cycles, C 2...

let $m$ be an $r$-module and $0 neq fin m^*={rm hom}(m,r)$. we associate an undirected graph $gf$ to $m$ in which non-zero elements $x$ and $y$ of $m$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. weobserve that over a commutative ring $r$, $gf$ is connected anddiam$(gf)leq 3$. moreover, if $gamma (m)$ contains a cycle,then $mbox{gr}(gf)leq 4$. furthermore if $|gf|geq 1$, then$gf$ is finit...

Suppose S is a subset of a metric spaceM with a metric δ, and D a subset of positive real numbers. The distance graph G(S, D), with a distance set D, is the graph with vertex set S in which two vertices x and y are adjacent iff δ(x, y) ∈ D. Distance graphs, first studied by Eggleton et al. [7], were motivated by the well-known plane-coloring problem: What is the minimum number of colors needed ...

Let χ(G) and χf (G) denote the chromatic and fractional chromatic numbers of a graph G, and let (n+, n0, n−) denote the inertia of G. We prove that: 1 + max ( n+ n− , n− n+ ) 6 χ(G) and conjecture that 1 + max ( n+ n− , n− n+ ) 6 χf (G). We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with...

Given a weighted graph Gx, where (x(v) : v ∈ V ) is a non-negative, realvalued weight assigned to the vertices of G, let B(Gx) be an upper bound on the fractional chromatic number of the weighted graph Gx; so χf (Gx) ≤ B(Gx). To investigate the worst-case performance of the upper bound B, we study the graph invariant β(G) = sup x 6=0 B(Gx) χf (Gx) . In recent work a particular upper bound resul...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...