Jenő Lehel

Toughness, hamiltonicity and split graphs

Abstract Related to Chvatals famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonicity on special classes of graphs. First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences. Furthermore, we present a polynomial time algorithm deciding whether the toughness of a given split graph is less than one and show that deciding whether the toughness of a bipartite graph is exactly one is coNP-complete. We show that every 3 2 - tough split graph is hamiltonian and that there is a sequence of non-hamiltonian split graphs with toughness converging to 3 2 .

An Asymptotic Ratio in the Complete Binary Tree

AbstractLet Tn be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into Tn as follows A(n;T)=|{S

Linear sets with five distinct differences among any four elements

\subseteq

Edge list multicoloring trees: An extension of Hall's theorem: EDGE LIST MULTICOLORING TREES

Tn : 1n∈S, S≅T}|, and B(n;T)=|{S

Even Cycles in Graphs with Many Odd Cycles

\subseteq

The path -pairability number of products of stars

Tn:1n∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim n→∞[A(n;T)/B(n;T)]=2ℓ;−1−1, for any tree T with ℓ leaves.