In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The ...

A cluster of cycles (or (r; q)-polycycle) is a simple planar 2{connected nite or countable graph G of girth r and maximal vertex-degree q, which admits an (r; q)-polycyclic realization P(G) on the plane. An (r; q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r) are combinatorial r-g...

This paper describes an efficient algorithm to find a Hamiltonian circuit in an arbitrary 4-connected maximal planar graph. The algorithm is based on our simlplified version of Whitney's proof of his theorem: every 4-connected maximal planar graph has a Hamiltonian circuit.

We will examine the maximal number of edges of a graph on p vertices of order dimension 4. We will show that the lower bound for this number is greater than 3 8 p 2 + 2p − 13. In particular the Turán-4 graph on p vertices does not have the maximal number of edges among the graphs of order dimension 4.

We introduced here an interesting tool for the structural study of chordal graphs, namely the Reduced Clique Graph. Using some of its combinatorial properties we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal under inclusion among all minimal separators.

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uni...

Let cl(G) denote the Ryjáček’s closure of a claw-free graph G. In this paper, we prove the following result. Let G be a 4-connected claw-free graph. Assume that G[NG(T )] is cyclically 3-connected if T is a maximal K3 in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.’s theorem [J. Graph Theory 48 (2005), no. 4, 267-276] and Pfend...

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge c...

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group Γ = 〈 x, y | x = y = 1 〉 of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the (2, 3, 7)-triangle group in the context...

The problem of computing all maximal induced subgraphs of a graph G that have a graph property P , also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary graph properties. The maximal Psubgraphs problem is reduced to restricted versions of this problem by providing algorithms that solve the general problem, ...