A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a conjecture of Chvatal ...

We deal with different algorithmic questions regarding properly arc-colored s-t trails, paths and circuits in arc-colored digraphs. Given an arc-colored digraph D with c ≥ 2 colors, we show that the problem of determining the maximum number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of a properly arc-colored s-...

—The n-dimensional augmented cube, denoted as AQ n , a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQ n lies on a fault-free cycle of every length from 4 to 2 n , even if there are up to 2n  3 link faults. We also show that this result is optimal. I. INTRODUCTION A graph G is a triple consisting of a vert...

In 4] it is conjectured that all diregular c-partite tournaments, with c 4, are pan-cyclic. In this paper we show that all diregular c-partite tournaments, with c 5, are in fact vertex-pancyclic.

Isaak posed the following problem. Suppose T is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in T equals the cardinality of minimum feedback arc set of T? We prove that the answer to the problem is in the negative. Further, we study the number of arc-disjoint cycles through a ve...

If D is a strongly connected digraph, then an arc set S of D is called a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D − V (D1) contains an arc. Recently, Volkmann [12] defined the restricted arc-connectivity λ(D) as the minimum cardinality over all restricted arc-cuts S. A strongly connected digraph D is called λconnected when λ(D) exists. Let k ≥ 2 be an i...