A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by χk(G). We determine the exact value of this parameter when G is a web or an antiweb. Our r...

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by χk(G). We determine the exact value of this parameter when G is a web or an antiweb. Our r...

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.

Given a wireless network where some pairs of communication links interfere with each other, we study sufficient conditions for determining whether a given set of minimum bandwidth quality-of-service (QoS) requirements can be satisfied. We are especially interested in algorithms which have low communication overhead and low processing complexity. The interference in the network is modeled using ...

We prove that every subcubic triangle-free graph has fractional chromatic number at most 14/5, thus confirming a conjecture of Heckman and Thomas [A new proof of the independence ratio of triangle-free cubic graphs. Discrete Math. 233 (2001), 233–237].

The concepts of (k; d)-coloring and the star chromatic number, studied by Vince, by Bondy and Hell, and by Zhu are shown to reeect the cographic instance of a wider concept, that of fractional nowhere-zero ows in regular matroids.

Let $\Gamma$ be a countable group and let $G$ the Schreier graph of free part Bernoulli shift $\Gamma$. We show that Borel fractional chromatic number is equal to $1$ over measurable independence $G$. As consequence, we asymptotically determine when group, answering question Meehan.