Gyula Y. Katona

Hamiltonian chains in hypergraphs

A graph G is said to be Pt-free if it does not contain an induced path on t vertices. The i-center Ci(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t-2⌋ ≤ i ≤ t - 2, with the property that, in every connected Pt-free graph G, the i-center Ci(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[Ci(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently.

Extremal k-edge-hamiltonian hypergraphs

An r-uniform hypergraph is k-edge-hamiltonian iff it still contains a hamiltonian-chain after deleting any k edges of the hypergraph. What is the minimum number of edges in such a hypergraph? We give lower and upper bounds for this question for several values of r and k.

Packing paths of length at least two

We give a simple proof for Kanekos theorem which gives a sufficient and necessary condition for the existence of vertex disjoint paths in a graph, each of length at least two, that altogether cover all vertices of the original graph. Moreover we generalize this theorem and give a formula for the maximum number of vertices that can be covered by such a path system.

Odd subgraphs and matchings

Let G be a graph and f : V(G) → {1, 3, 5,...}. Then a subgraph H of G is called a (1, f)-odd subgraph if degH(x) ∈ {1,3, ..., f(x)} for all x ∈ V(H). If f(x) = 1 for all x ∈ V(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K| > |H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.

Hypergraph Extensions of the Erdős-Gallai Theorem

Abstract The Erdős-Gallai Theorem gives the maximum number of edges in a graph without a path of length k. We extend this result for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Chrodality and 2-factors in tough graphs

A graph G is chordal if it contains no chordless cycle of length at least four and is k-chordal if a longest chordless cycle in G has length at most k. In this note it is proved that all ..-tough 5-chordal graphs have a 2-factor. This result is best possible in two ways. Examples due to Chvatal show that for all e>0 there exists a (..-e)-tough chordal graph with no 2-factor. Furthermore, examples due to Bauer and Schmeichel show that the result is false for 6-chordal graphs.

Toughness and edge-toughness

We generalize the definition of toughness and define a new property of graphs, the egdetoughness. In this new definition edges are also allowed to be deleted besides vertices. A number of results are given on the relation of toughness and edge-toughness.

Structure theorem and algorithm on (1,f)-odd subgraph

The authors give a Gallai-Edmonds type structure theorem on (1,f)-odd subgraphs and a polynomial algorithm for finding an optimal (1,f)-odd subgraph. Lovasz [The factorization of graphs. II. Acta Math. Acad. Sci. Hungar. 23 (1972) 223-246] and Cornuejols [General factors of graphs, J. Combin. Theory Ser. B 45(2) (1988) 185-198] solved these problems for a more general problem, the general factor problem with gaps at most 1. However, the statements of the theorems and the algorithm are much more simple in this special case, so it is worth of interest on its own. Also, the algorithm given for this case is faster than the general algorithm. The proofs follow a direct approach instead of deducing from the general case.

Breaking the hierarchy - a new cluster selection mechanism for hierarchical clustering methods

BackgroundHierarchical clustering methods like Wards method have been used since decades to understand biological and chemical data sets. In order to get a partition of the data set, it is necessary to choose an optimal level of the hierarchy by a so-called level selection algorithm. In 2005, a new kind of hierarchical clustering method was introduced by Palla et al. that differs in two ways from Wards method: it can be used on data on which no full similarity matrix is defined and it can produce overlapping clusters, i.e., allow for multiple membership of items in clusters. These features are optimal for biological and chemical data sets but until now no level selection algorithm has been published for this method.ResultsIn this article we provide a general selection scheme, the level independent clustering selection method, called LInCS. With it, clusters can be selected from any level in quadratic time with respect to the number of clusters. Since hierarchically clustered data is not necessarily associated with a similarity measure, the selection is based on a graph theoretic notion of cohesive clusters. We present results of our method on two data sets, a set of drug like molecules and set of protein-protein interaction (PPI) data. In both cases the method provides a clustering with very good sensitivity and specificity values according to a given reference clustering. Moreover, we can show for the PPI data set that our graph theoretic cohesiveness measure indeed chooses biologically homogeneous clusters and disregards inhomogeneous ones in most cases. We finally discuss how the method can be generalized to other hierarchical clustering methods to allow for a level independent cluster selection.ConclusionUsing our new cluster selection method together with the method by Palla et al. provides a new interesting clustering mechanism that allows to compute overlapping clusters, which is especially valuable for biological and chemical data sets.

Extremal P 4 -stable graphs

Abstract We call a graph G k -stable (with respect to some graph H ) if, deleting any k edges of G , the remaining graph still contains H as a subgraph. For a fixed H , the minimum number of edges in a k -stable graph is denoted by S ( k ) . We prove general bounds on S ( k ) and compute the exact value of the function S ( k ) for H = P 4 . The main result can be applied to extremal k -edge-Hamiltonian hypergraphs.