A note on interval edge-colorings of graphs

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an interval $t$-coloring if for each $i\in {1,2,\ldots,t}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this paper we prove that if a connected graph $G$ with $n$ vertices admits an interval $t$-coloring, then $t\leq 2n-3$. We also show that if $G$ is a connected $r$-regular graph with $n$ vertices has an interval $t$-coloring and $n\geq 2r+2$, then this upper bound can be improved to $2n-5$.

💡 Research Summary

The paper investigates interval edge‑colorings, a special type of proper edge‑coloring where the colors incident to each vertex form a consecutive set of integers and every color from 1 to t appears on at least one edge. A graph that admits such a coloring is called interval‑colorable, and for a given graph G the largest integer t for which an interval t‑coloring exists is denoted W(G).

The authors first recall several known results. Theorem 1 (Asratian‑Kamalian) states that any connected triangle‑free (in particular, any connected bipartite) interval‑colorable graph satisfies W(G) ≤ |V(G)| − 1. For certain families of bipartite graphs this bound can be sharpened; Theorem 2 (Asratian‑Casselgren) shows that a connected (a,b)‑biregular bipartite graph with at least 2(a+b) vertices satisfies W(G) ≤ |V(G)| − 3. For general graphs Kamalian proved Theorem 3: W(G) ≤ 2|V(G)| − 3, later improved to W(G) ≤ 2|V(G)| − 4 (Theorem 4). On the other hand, Theorem 5 (Petrosyan) demonstrates that the bound 2|V(G)| is essentially tight: for any ε > 0 there exists a graph with W(G) ≥ (2 − ε)|V(G)|. For planar graphs a stronger bound W(G) ≤ (11/6)|V(G)| is known (Theorem 6).

The main contribution of the paper is a short, conceptually clear proof of Theorem 3 based on the bipartite bound of Theorem 1, and a new improved bound for regular graphs.

Proof of Theorem 3 (short version).
Given a connected graph G with vertices v₁,…,v_n and an interval W(G)‑coloring α, the authors construct an auxiliary bipartite graph H. The vertex set of H consists of two copies of V(G): U = {u₁,…,u_n} and W = {w₁,…,w_n}. For each edge v_i v_j ∈ E(G) they add the two edges u_i w_j and u_j w_i to H, and they also add a “matching” edge u_i w_i for every i. Thus |V(H)| = 2n and H is connected and bipartite.

A new edge‑coloring β of H is defined as follows: for each original edge v_i v_j, assign color α(v_i v_j)+1 to both u_i w_j and u_j w_i; for each matching edge u_i w_i, assign color max S(v_i,α)+2. Consequently β uses colors 2,…,W(G)+2, and the smallest color incident to each u_i or w_i is 2. By recoloring a single edge with color 1, β becomes an interval (W(G)+2)‑coloring of H. Since H is a connected bipartite graph belonging to N, Theorem 1 yields W(G)+2 ≤ |V(H)| − 1 = 2n − 1, i.e. W(G) ≤ 2n − 3.

Improved bound for regular graphs (Theorem 7).
If G is r‑regular, connected, with n ≥ 2r+2, the same construction produces a bipartite graph H that is (r+1)‑regular. Because |V(H)| ≥ 2(2r+2), Theorem 2 (the biregular bound) applies, giving W(G)+2 ≤ |V(H)| − 3 = 2n − 3 and therefore W(G) ≤ 2n − 5.

Thus the paper provides a streamlined proof of the classic 2|V|‑3 upper bound for interval‑colorable graphs and refines it to 2|V|‑5 for sufficiently large regular graphs. These results tighten our understanding of how the structure of a graph (bipartiteness, regularity, size) constrains the maximum number of colors in an interval edge‑coloring, and they may guide future work on exact values of W(G) for specific graph families.