The matching preclusion number of a graph is the minimum number of neither edges whose deletion in a graph has a neither perfect matching nor an almost perfect matching. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently the conditional matching preclusion number of a graph was introduced to look for sets beyond those induced by a single...

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a speciic innnite sequence of 3-colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that if a (not necessarily maximal) triangle-free graph of order n has minimum degree n=...

The height of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let h(t) and jtj stand respectively for the height and the length of a trace t. We prove that the bivariate commutative series P t x h(t) y jtj is rational, and we give a nite representation of it. We use this result to ob...

The problem of determining if the on-line chromatic number of a graph is less than or equal to k, given a pre-coloring, is shown to be PSPACE-complete.

We study Abelian sandpiles on graphs of the form G×I, where G is an arbitrary finite connected graph, and I ⊂ Z is a finite interval. We show that for any fixed G with at least two vertices, the stationary measures μI = μG×I have two extremal weak limit points as I ↑ Z. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show ...

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr&minus1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are co...

We find new facet-defining inequalities for the linear ordering polytope generalizing the well-known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1 2 }-cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have ...

A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.