Interval edge-colorings of complete graphs

Abstract

An edge-coloring of a graph G with colors 1, 2, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. For an interval colorable graph G, W (G) denotes the greatest value of t for which G has an interval t-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of W (K 2n ) is known only for n ≤ 4. The second author showed that if n = p2 q , where p is odd and q is nonnegative, thenwhere n 2 is the number of 1's in the binary representation of n.In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on W (K 2n ) and determine its exact values for n ≤ 12.

On improper interval edge colourings

“…Finally, we discuss the improper interval colourings of complete graphs, which is of particular interest due to the ongoing intensive research in [9], [10]. Let z be a vertex of K n such that zx has colour 2.…”

“…let t(G) denote the maximum number of colours used in any interval colouring of G. The notion of interval colouring was introduced by Asratian and Kamalian in [1] in connection with specialized scheduling problems and, since then, it was further investigated in many papers, see for example [3], [5], [6], [7], [8], [9], [10]. Not all graphs are proper interval colourable (this concerns, for example, graphs of Class 2); in fact, the problem of determining whether a graph has an interval colouring is NP-complete, even for bipartite graphs, see [2] and [11].…”

Interval Edge-Coloring of Complete and Complete Bipartite Graphs With Restrictions

“…Interval edge-colorings have been intensively studied in different papers. Lower and upper bounds on the number of colors in interval edge-colorings were provided in [3,4] and the bounds were improved for different graphs: planar graphs [5], r-regular graphs with at least 2 ⋅ 𝑟 + 2 vertices [6], cycles, trees, complete bipartite graphs [3], n-dimensional cubes [7,8], complete graphs [9,10], Harary graphs [11], complete k-partite graphs [12], even block graphs [13]. In [14], interval edge-colorings with restrictions on edges were considered.…”

Interval Edge-Colorings of Trees With Restrictions on the Edges

“…Interval edge-colorings have been intensively studied in different papers. In [3] it was shown that every tree is from N. Lower and upper bounds on the number of colors in interval edge-colorings were provided in [4] and the bounds were improved for different graphs: planar graphs [5], r-regular graphs with at least 2r + 2 vertices [6], cycles, trees, complete bipartite graphs [3], n-dimensional cubes [7,8], complete graphs [9,10], Harary graphs [11], complete k-parite graphs [12].…”