Toughness, hamiltonicity and split graphs

Related to Chvátal's 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 split graph is hamiltonian and that there is a sequence of non-hamiltonian split graphs with toughness converging to .     

Alternating cycle-free matchings

The investigation of alternating cycle-free matchings is motivated by the Jump-number problem for partially ordered sets and the problem of counting maximum cardinality matchings in hexagonal systems.We show that the problem of deciding whether a given chordal bipartite graph has an alternating cycle-free matching of a given cardinality is NP-complete. A weaker result, for bipartite graphs only, has been known for some time. Also, the alternating cycle-free matching problem remains NP-complete for strongly chordal split graphs of diameter 2.In contrast, we give algorithms to solve the alternating cycle-free matching problem in polynomial time for bipartite distance hereditary graphs (time O(m ²) on graphs with m edges) and distance hereditary graphs (time O(m ⁵)).