Edge precoloring extension of trees II

Abstract

We consider the problem of extending and avoiding partial edge colorings of trees; that is, given a partial edge coloring ϕ of a tree T we are interested in whether there is a proper ∆(T )-edge coloring of T that agrees with the coloring ϕ on every edge that is colored under ϕ; or, similarly, if there is a proper ∆(T )-edge coloring that disagrees with ϕ on every edge that is colored under ϕ. We characterize which partial edge colorings with at most ∆(T ) + 1 precolored edges in a tree T are extendable, thereby proving an analogue of a result by Andersen for Latin squares. Furthermore we obtain some "mixed" results on extending a partial edge coloring subject to the condition that the extension should avoid a given partial edge coloring; in particular, for all 0 ≤ k ≤ ∆(T ), we characterize for which configurations consisting of a partial coloring ϕ of ∆(T ) − k edges and a partial coloring ψ of k + 1 edges of a tree T , there is an extension of ϕ that avoids ψ.

Extending partial edge colorings of cartesian products of graphs

“…A color c appears at a vertex v if some edge incident with v is colored c; otherwise c is missing at v. Let us first consider trees. In [6], it was proved that if at most ∆(T ) edges are properly precolored in a tree T , then the partial coloring ϕ can be extended to a proper ∆(T )-edge-coloring of T unless ϕ satisfies any of the following conditions: (C1) there is an uncolored edge uv in T such that u is incident with edges of k < ∆(T ) distinct colors and v is incident to ∆(T ) − k edges colored with ∆(T ) − k other distinct colors (so uv is adjacent to edges of ∆(T ) distinct colors);…”

“…The set of all colorings of T with ∆(T ) ≥ 2 satisfying the corresponding conditions above are denoted by C i for i = 1, 2, 3, 4 and we define C = ∪C i . We state the result of [6] as the following theorem.…”

“…Theorem 2.1. [6] Let T be a forest with maximum degree ∆(T ). If ϕ is a ∆(T )-edge precoloring of T with at most ∆(T ) precolored edges and ϕ / ∈ C, then ϕ is extendable to a proper ∆(T )-edge coloring of T .…”

“…In particular, every partial d-edge coloring with at most d − 1 colored edges is extendable to a d-edge coloring of Q d . This line of investigation was continued in [6,7] where similar questions are investigated for trees.…”

Extending partial edge colorings of Cartesian products of graphs

“…In particular, every partial coloring with at most d − 1 colored edges is extendable to a d-edge coloring of Q d . This line of investigation was continued in [6,7] where Extending Partial Edge Colorings of Cartesian Products 3 similar questions are investigated for trees.…”