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 a vertex‑centric variant of interval edge‑coloring. For a simple graph G, a proper edge t‑coloring assigns colors 1,…,t to edges so that adjacent edges receive different colors and every color is used at least once. The set of colors incident to a vertex x, denoted S(x), is called the spectrum of x. If S(x) forms a consecutive integer interval, x is an “interval vertex”; if the interval starts at 1, x is a “persistent‑interval vertex”. The authors introduce two parameters:
- μ_int(G,t): the maximum, over all proper edge t‑colorings, of the minimum number of interval vertices guaranteed;
- μ_pint(G,t): the analogous quantity for persistent‑interval vertices.

The main contribution is a collection of lower and upper bounds for these parameters across several well‑studied graph families, together with general inequalities that depend only on basic degree information.

Regular graphs. For a Δ‑regular graph G and any t ≥ Δ, the authors prove
 μ_int(G,t) ≥ ⌈|V(G)|·Δ/t⌉ and μ_pint(G,t) ≥ ⌈|V(G)|·Δ/t⌉.
Thus, when the number of colors is close to the degree, almost all vertices can be made interval (or persistent‑interval). They also exhibit constructions showing that for t = Δ+1 the bound is tight.

Bipartite graphs. Let G be bipartite with partite sets of sizes m ≤ n. For the complete bipartite graph K_{m,n} the minimal feasible number of colors is m + n − 1; at this value every vertex is persistent‑interval, i.e. μ_pint(K_{m,n},m+n−1)=m+n. When t exceeds this minimum, the guaranteed number of (persistent‑)interval vertices drops to at least m, independent of t. Similar bounds are derived for arbitrary bipartite graphs using Hall’s marriage theorem.

Trees. For a tree T with maximum degree Δ, minimum degree δ, and L leaves, the paper shows μ_int(T,t) ≥ L for any t ≥ Δ, because each leaf automatically has a one‑element spectrum, which is an interval. Moreover, if t ≥ 2Δ − 1, a constructive algorithm yields a coloring in which every vertex becomes interval, and for star graphs K_{1,Δ} the bound μ_pint(K_{1,Δ},Δ)=Δ+1 holds.

Complete graphs. For the complete graph K_n, known results on interval edge‑colorings give that when n is even and t = n − 1 all vertices are interval. When t = n, the authors prove an upper bound μ_pint(K_n,n) ≤ ⌈n/2⌉, showing that adding a single extra color can halve the number of vertices that retain a persistent‑interval spectrum.

General graphs. Combining the above, the authors establish universal inequalities:
 μ_int(G,t) ≥ |V(G)|·δ(G)/t, μ_pint(G,t) ≥ |V(G)|·δ(G)/t,
where δ(G) is the minimum degree. Conversely, for t = Δ(G)+k (k ≥ 1) they obtain upper bounds of the form μ_int(G,t) ≤ |V(G)|·Δ(G)/(Δ(G)+k). The proofs rely on matching theory (including König’s theorem for bipartite graphs), parity arguments, and a “color compression” technique that iteratively modifies a given coloring to increase the number of interval vertices without violating properness.

The paper emphasizes that these vertex‑level interval measures are distinct from the classical question of whether a graph admits a global interval edge‑coloring. Instead, μ_int and μ_pint quantify how many vertices can be satisfied simultaneously, which is directly relevant to applications such as time‑slot scheduling, frequency assignment, and parallel processing where only a subset of resources needs contiguous allocations.

Finally, the authors outline future directions: (i) determining exact values of μ_int and μ_pint for broader graph families, (ii) extending the framework to multigraphs or to edge‑colorings with additional constraints (e.g., list‑colorings), and (iii) developing polynomial‑time algorithms that, given (G,t), construct a coloring achieving the proven lower bounds.