Interference patterns in bijective colorings of 2-regular graphs Original Research Article

A bijective coloring of a simple graph of order n is a map from the vertex set of the graph onto {1,2,…,n}. Let f be a bijective coloring of G=(V,E) with |V|=n, and let α denote an interference parameter in {0,1,2,…,n−1}. The interference number of f with respect to G and α is the number of edges {u,v}∈E for which min{|f(u)−f(v)|, n−|f(u)−f(v)|}⩽α. We address two interference number problems for the family G2(n) of all 2-regular graphs of order n. For any given (n,α) with 0⩽α⩽n−1, the first problem is to determine a G∈G2(n) and a bijective coloring of G whose interference number is as small as possible. Given (n,α), the second problem is to determine a G∈G2(n) whose minimum interference number over all bijective colorings of G is as large as possible. Complete solutions are provided for both problems.