Longest cycles in threshold graphs

AbstractThe length of a longest cycle in a threshold graph is obtained in terms of a largest matching in a specially structured bipartite graph. It can be computed in linear time. As a corollary, Hamiltonian threshold graphs are characterized. This characterization yields Golumbic's characterization and sharpens Minty's characterization. It is also shown that a threshold graph has cycles of length 3, …, l where l is the length of a longest cycle

A note on perfect orders

AbstractPerfectly orderable graphs were introduced by Chvátal in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplical vertex, in the graph or in its complement. Second, weprovide a characterization of graphs G with this property: each maximal vertex ofG is simplical in the complement of G. Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order

Stability in CAN-free graphs

AbstractA class F of graphs characterized by three forbidden subgraphs C, A, N is considered; C is the claw (the unique graph with degree sequence (1, 1, 1, 3)), A is the antenna (a graph with degree sequence (1, 2, 2, 3, 3, 3) which does not contain C), and N is the net (the unique graph with degree sequence (1, 1, 1, 3, 3, 3)). These graphs are called CAN-free. A construction is described which associates with every CAN-free graph G another CAN-free graph G′ with strictly fewer nodes than G and with stbility number α(G′) = α(G) − 1. This gives a good algorithm for determining the stability number of CAN-free graphs

On simplicial and co-simplicial vertices in graphs

AbstractWe investigate the class of graphs defined by the property that every induced subgraph has a vertex which is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set). In particular we give the list of minimal forbidden subgraphs for the subclass of graphs whose vertex-set can be emptied out by first recursively eliminating simplicial vertices and then recursively eliminating co-simplicial vertices

Feasible edge colorings of trees with cardinality constraints

AbstractA variation of preemptive open shop scheduling corresponds to finding a feasible edge coloring in a bipartite multigraph with some requirements on the size of the different color classes. We show that for trees with fixed maximum degree, one can find in polynomial time an edge k-coloring where for i=1,…,k the number of edges of color i is exactly a given number hi, and each edge e gets its color from a set ϕ(e) of feasible colors, if such a coloring exists. This problem is NP-complete for general bipartite multigraphs. Applications to open shop problems with costs for using colors are described