A note on upper bounds for the maximum span in interval edge colorings of graphs

An edge coloring of a graph $G$ with colors $1,2,…, t$ is called an interval $t$-coloring if for each $i\in {1,2,…,t}$ there is at least one edge of $G$ colored by $i$, the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In 1994 Asratian and Kamalian proved that if a connected graph $G$ admits an interval $t$-coloring, then $t\leq (d+1) (\Delta -1) +1$, and if $G$ is also bipartite, then this upper bound can be improved to $t\leq d(\Delta -1) +1$, where $\Delta$ is the maximum degree in $G$ and $d$ is the diameter of $G$. In this paper we show that these upper bounds can not be significantly improved.

💡 Research Summary

The paper investigates the relationship between the span of an interval edge‑coloring and two fundamental graph parameters: the diameter d and the maximum degree Δ. An interval t‑coloring of a graph G uses colors 1,…,t, each color appears on at least one edge, and for every vertex the incident edge colors are distinct and form a consecutive integer interval. Asratian and Kamalian (1994) proved that if a connected graph G admits an interval t‑coloring then
 t ≤ (d + 1)(Δ − 1) + 1,
and if G is bipartite the bound improves to
 t ≤ d(Δ − 1) + 1.
The present work shows that these bounds are essentially tight; they cannot be significantly lowered.

The authors construct two families of graphs that attain (or come arbitrarily close to) the above limits.

Theorem 5 (general graphs).
For any integers d ≥ 1 and q ≥ 1 they define
 G_{d,q} = P_d □ K_{2q},
the Cartesian product of a path of length d and a complete graph on 2q vertices. G_{d,q} is connected, has diameter d, and its maximum degree is
 Δ(G_{d,q}) = 2q − 1 (d = 1), 2q (d = 2), 2q + 1 (d ≥ 3).
Since both factors belong to the class N of interval‑colorable graphs, the product also belongs to N (Theorem 3). Using a known interval coloring of K_{2q} (Theorem 2) and shifting its colors by multiples of 2q for each layer, the authors produce an interval coloring of G_{d,q} that uses exactly
 t = (d + 1)·2q − 2 − q
colors. Substituting the appropriate Δ gives the lower bounds stated in Theorem 5, which are only q colors short of the general upper bound (d + 1)(Δ − 1) + 1. Hence the bound cannot be improved by more than a constant independent of d and Δ.

Theorem 6 (bipartite graphs).
For any integers d ≥ 1, Δ ≥ 2 the authors construct a connected bipartite graph G_{d,Δ} with diameter d and maximum degree Δ such that
 W(G_{d,Δ}) = d(Δ − 1) + 1,
exactly matching the bipartite upper bound. The construction is case‑based:

• If Δ = 2 (any d ≥ 2) they use the even cycle C_{2d}, which is interval‑colorable with span d + 1 = d(Δ − 1) + 1.
• If d = 2 (any Δ ≥ 2) they use the complete bipartite graph K_{Δ,Δ}, whose span is 2Δ − 1 = d(Δ − 1) + 1.
• For d, Δ ≥ 3 they distinguish even and odd d.
  – When d is even, they build ⌊d/2⌋ “blocks”, each a copy of K_{Δ,Δ} minus a single edge, linked together through the vertices of maximum degree. An explicit coloring assigns to edges inside block i the color (i − 1)(2Δ − 1) + j + k − i, and to the inter‑block edges the color (i − 1)(2Δ − 1) − i + 2. This yields a consecutive interval of length d(Δ − 1) + 1.
  – When d is odd, the case d = 3 is handled by two copies of K_{Δ,Δ} plus a perfect matching between the two partite sets, colored in a straightforward way to achieve span 3Δ − 2 = d(Δ − 1) + 1. For d ≥ 5 they introduce auxiliary vertices a, c and additional pendant vertices to keep the diameter d while preserving regularity. A more intricate coloring λ is defined, again producing exactly d(Δ − 1) + 1 colors.

Thus for every admissible pair (d, Δ) a bipartite graph attaining the bound exists, proving that the bound in Theorem 4(2) is tight.

The paper also relies on known results: Theorem 1 gives the exact span of K_{Δ,Δ}, Theorem 2 provides a lower bound for the span of complete graphs, and Theorem 3 guarantees that the Cartesian product of interval‑colorable graphs remains interval‑colorable. These tools enable the authors to construct the extremal examples systematically.

Implications.
The results settle a natural question about the optimality of the Asratian‑Kamalian bounds. They demonstrate that any attempt to lower the general bound (d + 1)(Δ − 1) + 1 or the bipartite bound d(Δ − 1) + 1 would contradict the existence of the families presented. Consequently, the known bounds are essentially best possible, up to an additive constant that does not depend on the graph size. Moreover, the constructions illustrate how Cartesian products and carefully designed regular bipartite structures can be used to achieve extremal interval colorings, suggesting potential techniques for related scheduling and resource‑allocation problems where interval constraints arise.

In summary, the paper provides a definitive answer to the optimality of the classic diameter‑degree upper bounds for interval edge colorings, enriching the theory with explicit extremal constructions and confirming that the bounds are tight.