Interval Edge Colourings of Complete Graphs and n-cubes

For complete graphs and n-cubes bounds are found for the possible number of colours in an interval edge colourings.

💡 Research Summary

The paper investigates interval edge colourings of two important families of graphs: complete graphs with an even number of vertices, denoted K₂ₙ, and n‑dimensional hypercubes Qₙ. An interval t‑colouring of a graph G is a proper edge‑colouring using colours 1,…,t such that for every vertex x the set of colours incident to x forms a set of d_G(x) consecutive integers. A graph that admits at least one interval t‑colouring belongs to the class N; the smallest and largest feasible values of t are denoted w(G) and W(G), respectively.

The authors begin by recalling known results: for triangle‑free graphs W(G) ≤ |V(G)|‑1, for bipartite graphs W(G) ≤ d(G)(Δ(G)‑1)+1, and for general graphs W(G) ≤ 2|V(G)|‑3 (or 2|V(G)|‑4 when |V(G)|≥3). For regular graphs, a necessary and sufficient condition for belonging to N is χ′(G)=Δ(G); moreover, if G∈N then every integer t with Δ(G)≤t≤W(G) yields an interval t‑colouring. The decision problem “does a given regular graph belong to N?” is NP‑complete.

The first original contribution concerns K₂ₙ. It is known that odd complete graphs are not interval‑colourable, while χ′(K₂ₙ)=Δ(K₂ₙ)=2n‑1, so w(K₂ₙ)=2n‑1. Earlier work gave the lower bound W(K₂ₙ) ≥ 3n‑2. The authors improve this bound by writing n as p·2^q where p is odd and q≥0. They prove

  W(K₂ₙ) ≥ 4n‑2‑p‑q.

The proof is constructive and inductive. Starting from a known interval colouring of K₂ₘ, they embed K₂ₘ as an induced subgraph of K₄ₘ and extend the colouring by adding a new block of colours of size 4m‑1, carefully preserving the consecutive‑colour property at every vertex. Repeating this construction yields the stated bound for any n. As a corollary, for every integer t satisfying

  2n‑1 ≤ t ≤ 4n‑2‑p‑q

there exists an interval t‑colouring of K₂ₙ.

The second contribution deals with hypercubes. Since Qₙ is a regular bipartite graph with degree n, χ′(Qₙ)=Δ(Qₙ)=n, and therefore w(Qₙ)=n. The authors establish a quadratic lower bound for the maximal feasible colour number:

  W(Qₙ) ≥ n(n+1)/2.

The argument uses the product representation Qₙ = K₂ × Qₙ₋₁. Assuming an interval W(Qₙ₋₁)‑colouring of Qₙ₋₁, they create two disjoint copies of Qₙ₋₁, colour each with the original scheme, and colour the “cross‑edges” (those joining corresponding vertices of the two copies) by adding n to the colour of the original edge. This yields an interval (W(Qₙ₋₁)+n)‑colouring of Qₙ. By induction from Q₁ (which trivially has W(Q₁)=1) the quadratic bound follows. Consequently, for every integer t with

  n ≤ t ≤ n(n+1)/2

the hypercube Qₙ admits an interval t‑colouring.

The paper also notes that odd complete graphs K_p (p odd) are not in N, and reiterates the NP‑completeness of determining interval colourability for regular graphs. Overall, the work extends the theory of interval edge colourings by providing explicit constructive lower bounds for two fundamental graph families, thereby narrowing the gap between known upper and lower limits and offering concrete colour‑assignment algorithms for a wide range of colour numbers.