Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at
least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian …

In light of the September 11, 2001, terrorist attacks, an abundance of network security issues
arose and immediately came to the forefront of mathematical research. One such notion of …

An edge-colored directed graph is called properly connected if, between every pair of
vertices, there is a properly colored directed path. We study some conditions on directed …

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path
are colored with one same color. An edge-colored graph is proper connected if any two …

A proper edge-coloring of a graph is a coloring in which adjacent edges receive distinct
colors. A path is properly colored if consecutive edges have distinct colors, and an edge …

A coloring of a graph G is properly connected if every two vertices of G are the ends of a
properly colored path. We study the complexity of computing the proper connection number …

The (directed) proper connection number of a given (di) graph G is the least number of
colors needed to edge-color G such that there exists a properly colored (di) path between …

This note introduces the vertex proper connection number of a graph and provides a
relationship to the chromatic number of minimally connected subgraphs. Also a notion of …

A k-proper connected 2-coloring for a graph is an edge bipartition which ensures the
existence of at least k vertex disjoint simple alternating paths (ie, paths where no two …

A tree T in an edge-colored graph is a proper tree if any two adjacent edges of T are colored
with different colors. Let G be a graph of order n and k be a fixed integer with 2≤ k≤ n. For a …