Interval Colourings of Some Regular Graphs

Let G = (V,E) be an undirected graph without loops and multiple edges [1], V (G) and E(G) be the sets of vertices and edges of G, respectively. The degree of a vertex x ∈ V (G) is denoted by dG(x), the maximum degree of a vertex of G-by ∆(G), and the chromatic index [2] of G-by χ(G). A graph is regular, if all its vertices have the same degree. If α is a proper edge colouring of the graph G [3], then the color of an edge e ∈ E(G) in the colouring α is denoted by α(e,G), and by α(e) if from the context it is clear to which graph it refers. For a proper edge colouring α, the set of colors of the edges that are incident to a vertex x ∈ V (G), is denoted by S(x, α). A proper colouring α of edges of G with colors 1, 2, . . . , t is interval [4], if for each color i, 1 ≤ i ≤ t, there exists at least one edge ei ∈ E(G) with α(ei) = i and the edges incident with each vertex x ∈ V (G) are colored by dG(x) consecutive colors. For t ≥ 1 let Nt denote the set of graphs which have an interval t-colouring, and assume: N ≡ ⋃