Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,…,t$ such that each of $t$ colors is used, and adjacent edges are colored differently. The set of colors of edges incident with a vertex $x$ of $G$ is called a spectrum of $x$. A proper edge $t$-coloring of a graph $G$ is interval for its vertex $x$ if the spectrum of $x$ is an interval of integers. A proper edge $t$-coloring of a graph $G$ is persistent-interval for its vertex $x$ if the spectrum of $x$ is an interval of integers beginning from the color 1. For graphs $G$ from some classes of graphs, we obtain estimates for the possible number of vertices for which a proper edge $t$-coloring of $G$ can be interval or persistent-interval.

💡 Research Summary

The paper investigates how many vertices of a graph can have an “interval spectrum” or a “persistent‑interval spectrum” under a proper edge‑t‑coloring. For a proper edge coloring ϕ of a graph G, the set of colors incident to a vertex x, denoted S_G(x,ϕ), is called the spectrum of x. If S_G(x,ϕ) consists of d_G(x) consecutive integers, ϕ is said to be interval for x; if the interval starts at 1, ϕ is persistent‑interval for x. The authors define two parameters:
 η_i(G) = max_{ϕ∈α(G)} |{x∈V(G) : ϕ is interval for x}|,
 η_pi(G) = max_{ϕ∈α(G)} |{x∈V(G) : ϕ is persistent‑interval for x}|.
These quantities measure, over all possible proper edge colorings, the largest possible number of vertices that can enjoy the respective spectral property.

The paper first reviews known results: for regular graphs, η_i(G)=|V(G)| ⇔ G admits an interval coloring ⇔ χ′(G)=Δ(G). Deciding whether χ′(G)=Δ(G) for a regular graph is NP‑complete, which implies that determining η_i(G)=|V(G)| (or η_pi(G)=|V(G)|) is also NP‑complete.

The main contributions are lower bounds for η_i(G) and η_pi(G) in two families of graphs.

1. Regular graphs with χ′(G)=Δ(G)+1.
   – Theorem 1 shows that for any regular graph G with χ′(G)=Δ(G)+1, at least ⌈|V(G)|/(Δ+1) vertices can be made persistent‑interval. The proof selects a proper (Δ+1)‑coloring β, partitions the vertex set according to which color is missing from each vertex’s spectrum, and picks the largest part. By a simple recoloring (either swapping the missing color with Δ+1 or rotating colors modulo Δ+1) a persistent‑interval coloring on that part is obtained.
   – Corollary 1 specializes this to cubic graphs (Δ=3), guaranteeing at least |V(G)|/4 vertices with a persistent‑interval spectrum.
   – Theorem 2 gives a stronger bound for ordinary interval spectra: η_i(G)≥⌈|V(G)|/⌊(Δ+1)/2⌋⌉. The argument again uses the missing‑color partition, but now groups colors in pairs (or a single color when Δ is even) to obtain intervals of length d_G(x). A modular shift of colors yields an interval coloring on the selected vertex set.
   – Corollary 2 yields for cubic graphs η_i(G)≥|V(G)|/2.

2. Bipartite (k‑1, k)‑regular graphs.
   The paper studies bipartite graphs G with bipartition (X,Y) where every vertex of X has degree k‑1 and every vertex of Y has degree k (k≥3). Since χ′(G)=Δ(G)=k, the authors can work with proper k‑colorings.
   – Theorem 3 (cited) states that any bipartite graph admits an interval coloring on one side of the bipartition.
   – Theorem 4 (cited) shows that if degrees on X are not larger than those on Y, then Y can be colored persistently‑interval.
   – Theorem 5 extends this: for any vertex x₀∈X, there exists a proper k‑coloring that is persistent‑interval on {x₀}∪Y. The construction repeatedly swaps two colors along a maximal alternating path starting at x₀, gradually moving the missing colors into the interval