Interval Edge-Colorings of Trees With Restrictions on the Edges

Abstract

An edge-coloring of a graph $G$ with consecutive integers $c_1,\ldots,c_t$ is called an interval t-coloring, if all colors are used, and the colors of edges incident to any 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 $t$. In this paper, we consider the case, where there are restrictions on the edges of the tree and provide a polynomial algorithm for checking interval colorability that satisfies those restrictions.

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

“…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. In this case there can be restrictions on the edges for the allowed colors.…”

Interval Vertex-Colorings of Cactus Graphs With Restrictions on Vertices

“…The list coloring problem is NP-complete for complete bipartite graphs, and can be solved in polynomial time for block graphs [11]. Interval edge-colorings with restrictions were also considered in different papers [12][13][14][15].…”