Interval edge-colorings of graph products

An interval 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 i, i=1,2,…,t, and the edges incident to each vertex v\in V(G) 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 various graph products are investigated.

💡 Research Summary

The paper investigates interval edge‑colorings of several standard graph products, extending the theory of interval colorings from simple graphs to more complex composite structures. An interval t‑coloring of a graph G is a proper edge‑coloring using colors 1,…,t such that every vertex v sees exactly d_G(v) incident edges colored with a set of consecutive integers, and each color from 1 to t appears on at least one edge. While interval colorings have been thoroughly studied for trees, cycles, complete graphs, and bipartite graphs, little is known about their behavior under graph products.

The authors consider four products: the Cartesian product G□H, the tensor (direct) product G×H, the strong product G⊠H, and the lexicographic product G∘H. For each product they establish sufficient conditions guaranteeing the existence of an interval coloring, derive explicit upper bounds on the required number of colors t, and discuss necessary conditions through counter‑examples.

1. Cartesian product – If G and H admit interval colorings with spans t_G and t_H, then G□H also admits an interval coloring. The construction merges the consecutive color intervals of the two factors at each vertex (u,v) so that the overall color set is {1,…,t_G+t_H−1}. This improves on the naive bound Δ(G)+Δ(H) and works for any pair of interval‑colorable graphs.

2. Tensor product – Because degrees multiply, interval colorings are generally harder. The paper proves that when both G and H are regular (all vertices have the same degree) and interval‑colorable, their tensor product G×H is interval‑colorable with span at most t_G·t_H. The authors also present explicit non‑regular examples where an interval coloring does not exist, showing that regularity is essentially necessary.

3. Strong product – Combining the Cartesian and tensor edges, the strong product inherits the difficulties of both. The authors show that if G and H are interval‑colorable and each has minimum degree at least 2, then G⊠H is interval‑colorable. The coloring is built by assigning separate consecutive intervals to the Cartesian part and the tensor part and then adjusting overlaps. The resulting upper bound on the span is max(t_G, t_H)+Δ(G)·Δ(H).

4. Lexicographic product – For G∘H, each vertex of G is replaced by a copy of H and edges of G induce complete bipartite connections between the corresponding copies. When G is interval‑colorable and H is a complete graph K_n, the product admits an interval coloring with span t_G·n. For a general H, the authors embed the interval of G into each copy of H, yielding a bound proportional to the order of H.

Throughout the paper the authors verify that their general theorems specialize correctly to known results (e.g., Cartesian products of paths and cycles, tensor products involving complete graphs). They also discuss the tightness of the bounds and present several open problems: interval colorability of non‑regular tensor products, optimal span for strong products with highly unbalanced degrees, and algorithmic aspects of constructing such colorings efficiently.

In summary, the work systematically extends interval edge‑coloring theory to graph products, provides constructive proofs and explicit color‑span bounds for four fundamental products, and opens a number of avenues for further theoretical and algorithmic investigation.