最大度树的边着色对策$Δ=4$
Edge colouring Game on Trees with maximum degree $Δ=4$
摘要：
考虑下面的游戏。我们有一棵树$T$和两个玩家（比如说）Alice和Bob，他们交替地给树的一条边上色（使用$k$颜色之一）。如果树的所有边都被着色，则Alice获胜，否则Bob获胜。的树的博弈色指数是Alice有获胜策略的最小指数$k$。如果树中节点的最大度为$Δ$，则Erdos等证明了树中节点的最大度为$Δ$. [6]，表明游戏色指数至少为$Δ+1$。已知对于$Δ\neq 4$的所有值，该界都是紧的。在本文中，我们表明，对于$Δ=4$，即使鲍勃被允许跳过一步，爱丽丝总是可以选择一个边的颜色和赢得比赛的$k=Δ+1$。因此，最大度树$4$的对策色指数也是$5$。因此，对于所有$Δ\geq 2$，最大度$Δ$树的对策色指数是$Δ+1$。此外，可以对树进行预处理，以允许Alice在$O(1)$时间内挑选下一个要着色的边缘。独立感兴趣的结果是一个线性时间算法的在线边缘删除问题的树。
英文摘要：
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 $Δ$, Erdos et.al.[6], show that the game chromatic index is at least $Δ+1$. The bound is known to be tight for all values of $Δ\neq 4$. In this paper we show that for $Δ=4$, even if Bob is allowed to skip a move, Alice can always choose an edge to colour and win the game for $k=Δ+1$. Thus the game chromatic index of trees of maximum degree $4$ is also $5$. Hence, game chromatic index of trees of maximum degree $Δ$ is $Δ+1$ for all $Δ\geq 2$. 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 edge-deletion problem on trees.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
