Supermagic graphs with many odd degrees

Abstract

A graph G = (V,E) is called supermagic if there exists a bijection f : E → {1, 2, … , |E|} such that the weight of every vertex x ∈ V defined as the sum of labels f(xy) of all edges xy incident with x is equal to the same number m, called the supermagic constant. Recently, Kovář et al. affirmatively answered a question by Madaras about existence of supermagic graphs with arbitrarily many different degrees. Their construction provided graphs with all degrees even. Therefore, they asked if there exists a supermagic graph with d different odd degrees for any positive integer d. We answer this question in the affirmative by providing a construction based on the use of 3-dimensional magic rectangles.