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**publisher and metadata sources**).We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem… Continue Reading

A vertex set Y in a (hyper)graph is called k-independent if in the sub(hyper)-graph induced by Y every vertex is incident to less than k edges. We prove a lower bound for the maximum cardinality of a… Continue Reading

A Boolean formula in a conjunctive normal form is called a $(k,s)$ – formula if every clause contains exactly k variables and every variable occurs in at most s clauses. The $(k,s)$–${\text{SAT}}$… Continue Reading

Let F = {F1,…} be a given class of forbidden graphs. A graph G is called F-saturated if no Fi ∈ F is a subgraph of G but the addition of an arbitrary new edge gives a forbidden subgraph. In this… Continue Reading

Abstract This paper is the first article in a series devoted to the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some… Continue Reading

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$,… Continue Reading

The partition problem is one of the basic NP-complete problems. While an efficient heuristic for the optimization version, which is equivalent to minimizing the makespan on two identical machines, is… Continue Reading

Abstract We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, (ii) a connected trangle-free graph, (iii) a connected C 4 -free graph with n vertices… Continue Reading

We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further… Continue Reading

The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi… Continue Reading