Let G = (V,E). A set S ⊆ V is independent if no two vertices from S are adjacent, and by Ind(G) we mean the set of all independent sets of G. The number d (X) = |X| − |N(X)| is the difference of X ⊆ V , and A ∈ Ind(G) is critical if d(A) = max{d (I) : I ∈ Ind(G)} [7]. Let us recall the following definitions: ker(G) = ∩{S : S is a critical independent set} [5], core (G) = ∩{S : S is a maximum in...

The independence ratio i(G) of a graph G is the ratio of its independence number and the number of vertices. The ultimate categorical independence ratio of a graph G is defined as limk→∞ i(G×k), where G×k denotes the kth categorical power of G. This parameter was introduced by Brown, Nowakowski and Rall, who asked about its value for complete multipartite graphs. In this paper we determine the ...

It is proven that if G is a 3-cyclable graph on n vertices, with minimum degree δ and with a maximum independent set of cardinality α, then G contains a cycle of length at least min{n, 3δ− 3, n+ δ − α}.

Let G be a graph of sufficiently large order n, and let a and b be integers with 1 ≤ a ≤ b. Let h : E(G) → [0, 1] be a function. If a ≤ ∑x∈e h(e) ≤ b holds for any x ∈ V (G), then G[Fh] is called a fractional [a, b]-factor of G with indicator function h, where Fh = {e ∈ E(G) | h(e) > 0}. A graph G is fractional independent-set-deletable [a, b]-factor-critical (simply, fractional ID-[a, b]-facto...

In 1973, Deuber published his famous proof of Rado’s conjecture regarding partition regular sets. In his proof, he invented structures called (m, p, c)-sets and gave a partition theorem for them based on repeated applications of van der Waerden’s theorem on arithmetic progressions. In this paper, we give the complete proof of Deuber’s, however with the more recent parameter set proof of his par...

Yannakakis’ Clique versus Independent Set problem (CL− IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O(n), and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a p...

A cycle C in a graph G is said to be dominating if E(G−C) = ∅. Enomoto et al. showed that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 2, then every longest cycle is dominating. But it is unknown whether the condition on the independence number is sharp. In this paper, we show that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 1, then G has a longest cycle which...

A graphG is fractional ID-[a, b]-factor-critical ifG−I includes a fractional [a, b]-factor for every independent set I of G. In this paper, it is proved that if α(G) ≤ 4b(δ(G)−a+1) (a+1)2+4b , then G is fractional ID-[a, b]-factor-critical. Furthermore, it is shown that the result is best possible in some sense.

The Maximum Independent Set problem often arises in the following form: given n tasks which compete for the same resources, compute the largest set of tasks which can be performed at the same time without sharing resources. More generally, in case the tasks have weights (priorities), one may ask for the set of tasks of largest total weight which can be performed at the same time. This is called...

Let n(G) and α(G) be the order and the independence number of a graph G, repsectively. If G is bipartite graph, then it is well-known and easy to see that α(G) ≥ n(G) 2 . In this paper we present a constructive characterization of bipartite graphs G for which