A connected graph G is said to be k-cycle resonant if, for 16 t6 k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and su4cient conditions for a graph to be k-cycle resonant were given. Rece...

Given a graph G and an edge coloring C of G, an alternating cycle of G is such a cycle of G in which any adjacent edges have distinct colors. Let dc(v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, some color degree conditions for the existence of alternating cycles of length 3 or 4 are obtained. We al...

We consider impulsive excitation of a linear polar molecule by a plane polarized electromagnetic "half-cycle" pulse in the terahertz range. A rotational wave packet is created with angular momentum states of different parity. The time evolution of the wave packet corresponds to alternating molecular orientations with respect to the polarization axis of the field. This field-free time-dependent ...

Let K n denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let ∆(K n) denote the maximum number of edges of the same color incident with a vertex of K n. A properly colored cycle (path) in K n, that is, a cycle (path) in which adjacent edges have distinct colors is called an alternating cycle (path). Our note is inspired by the following conjecture by B. boll...

We consider a graph with colored edges. A trail (vertices may repeat but not edges) is called alternating when successive edges have different colors. Given a set of vertices called terminals, the alternating reachability problem is to find an alternating trail connecting distinct terminals, if one exists. A special case with two colors is searching for an augmenting path with respect to a give...

We prove that on prefix-recognizable graphs reachability is complete for deterministic exponential time matching the complexity of alternating reachability.

In this paper we consider an approach to solve the Hamiltonian strictly alternating cycle problem. This approach is based on constructing logical models for the problem.

We consider a graph with colored edges. A trail (vertices may repeat but not edges) is called alternating when successive edges have different colors. Given a set of vertices called terminals, the alternating reachability problem is to find a trail connecting distinct terminals, if one exists. A special case with two colors is searching for an augmenting path with respect to a given matching. I...

Springer numbers are analogs of Euler numbers for the group of signed permutations. Arnol’d showed that they count some objects called snakes, which generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial pro...