The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices

Fong, Wai Lam; Chan, Wai Hong; Nong, Ge

doi:10.1007/s10878-018-0277-7

Cited by 2 publications

(2 citation statements)

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“…Fong et.al. [8] show that the bound is true provided each degree 4 node has at most one other neighbour of degree 4. Andre [2], Chan and Nong [5] and Fong et.al [8] allow Bob to skip moves.…”

Section: Introductionmentioning

confidence: 90%

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Edge colouring Game on Trees with maximum degree $Δ=4$

Singh,

Saxena

2020

Preprint

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Consider the following game. We are given a tree T and two players (say) Alice and Bob who alternately colour an edge of a tree (using one of k colours). If all edges of the tree get coloured, then Alice wins else Bob wins. Game chromatic index of trees of is the smallest index k for which there is a winning strategy for Alice. If the maximum degree of a node in tree is ∆, Erdos et.al. [6], show that the game chromatic index is at least ∆ + 1. The bound is known to be tight for all values of ∆ = 4.In this paper we show that for ∆ = 4, even if Bob is allowed to skip a move, Alice can always choose an edge to colour and win the game for k = ∆ + 1. Thus the game chromatic index of trees of maximum degree 4 is also 5. Hence, game chromatic index of trees of maximum degree ∆ is ∆ + 1 for all ∆ ≥ 2.Moreover,the tree can be preprocessed to allow Alice to pick the next edge to colour in O(1) time.A result of independent interest is a linear time algorithm for on-line edgedeletion problem on trees.

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Section: Introductionmentioning

confidence: 90%

“…[8] show that the bound is true provided each degree 4 node has at most one other neighbour of degree 4. Andre [2], Chan and Nong [5] and Fong et.al [8] allow Bob to skip moves. Fong and Chan [9] study the problem using colours greater than the game chromatic index.…”

Section: Introductionmentioning

confidence: 90%