Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori

An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2…,t such that at least one edge of G is colored by color i,i=1,2…,t, and the edges incident with each vertex v are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge colorings of bipartite cylinders and bipartite tori are investigated.

💡 Research Summary

The paper investigates interval edge‑colorings of two families of bipartite graphs: the bipartite cylinder (C(m,n)=P_m\Box C_n) (the Cartesian product of a path on (m) vertices and a cycle on (n) vertices) and the bipartite torus (T(m,n)=C_m\Box C_n) (the Cartesian product of two cycles). An interval edge‑(t)-coloring of a graph (G) is a proper edge‑coloring using colors (1,2,\dots ,t) such that (i) each color appears on at least one edge, and (ii) for every vertex (v) the set of colors incident to (v) forms a set of (d_G(v)) consecutive integers. The class (\mathcal N) consists of all graphs admitting an interval edge‑coloring; for a graph (G\in\mathcal N) the smallest feasible (t) is denoted (w(G)) and the largest is denoted (W(G)).

Previous work (Asratian & Kamalian, 1994) established that for any bipartite graph (G\in\mathcal N) one has the general upper bound (W(G)\le d(G),\Delta(G)-1), where (d(G)) is the diameter and (\Delta(G)) the maximum degree. Moreover, for regular graphs the condition (\chi’(G)=\Delta(G)) is equivalent to membership in (\mathcal N), and if a graph belongs to (\mathcal N) then every integer between (w(G)) and (W(G)) is also realizable as a feasible number of colors.

The contribution of the present paper is to provide explicit lower bounds for (W(G)) for the two graph families mentioned above, thereby narrowing the gap between known upper bounds and constructive lower bounds.

Theorem 3 (Cylinder). For the bipartite cylinder (C(m,n)) the authors construct a coloring (\alpha) defined by simple linear formulas in the vertex coordinates ((i,j)). Six types of edges (horizontal, vertical, and the wrap‑around edges of the cycle) receive colors such as (\alpha(e)=i+2j-1) or (\alpha(e)=i+2j-2), with appropriate adjustments at the boundaries. They prove that:

1. Every color from (1) up to (t=3m+2n-2) appears on at least one edge.
2. For each vertex the incident edges receive exactly (d_{C(m,n)}(v)) consecutive colors. Thus (\alpha) is an interval edge‑(t)-coloring, establishing the lower bound \