Using the framework of kernelization we study whether efficient preprocessing schemes for the Treewidth problem can give provable bounds on the size of the processed instances. Assuming the ANDdistillation conjecture to hold, the standard parameterization of Treewidth does not have a kernel of polynomial size and thus instances (G, k) of the decision problem of Treewidth cannot be efficiently r...

Let FG(P ) be a functional defined on the set of all the probability distributions on the vertex set of a graph G. We say that G is symmetric with respect to FG(P ) if the uniform distribution on V (G) maximizes FG(P ). Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic numb...

Let Mk ⊆ N be a given set that consists of k noncontiguous integers. Define Exact-Mk-Colorability to be the problem of determining whether χ(G), the chromatic number of a given graph G, equals one of the k elements of the set Mk exactly. In 1987, Wagner [Wag87] proved that Exact-Mk-Colorability is BH2k(NP)-complete, where Mk = {6k + 1, 6k + 3, . . . , 8k − 1} and BH2k(NP) is the 2kth level of t...

Is there, for all large values of n, a graph on n vertices with no clique of size 5 log n, and no independent set of size 5 log n? Is there a graph with some number n of vertices, no cycles of length less than 100, and no independent sets of size greater than n/100? Note that such a graph has chromatic number at least 100. Is there, for all large values of n, a graph on n vertices with maximum ...

In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze–Kannan-regular. With a more refined version of these two local conditions we provide a deterministic algorithm that obtains a Frieze–Kanan-regular partition of any graph G in time O(|V (G)|).

the edge tenacity te(g) of a graph g is dened as:te(g) = min {[|x|+τ(g-x)]/[ω(g-x)-1]|x ⊆ e(g) and ω(g-x) > 1} where the minimum is taken over every edge-cutset x that separates g into ω(g - x) components, and by τ(g - x) we denote the order of a largest component of g. the objective of this paper is to determine this quantity for split graphs. let g = (z; i; e) be a noncomplete connected spli...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...