A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths.
图$G$的关联是一对$(v,e)$，其中$v$是$G$的顶点，$e$是与$v$关联的边。当$v = w$、或$e = f$、或$vw = e$或$f$时，$(v,e)$和$(w,f)$两个事件相邻。关联着色博弈[S. D. Andres，The incidence game chromatic number，Discrete Appl.Math.157（2009），1980-1987]是普通着色游戏的一种变体，其中两个玩家Alice和Bob使用给定数量的颜色交替地对图的发生率着色，使得相邻的发生率得到不同的颜色。如果整个图都是彩色的，那么Alice就赢了，否则Bob就赢了。图$G$的关联博弈色数$i_g(G)$是Alice在$G$上进行关联着色博弈时获胜策略的最小颜色数。Andres证明了$\lceil 3/2 Δ(G)\rceil \le $ i_g（G）\le 2Δ（G）+ 4k - 2 $ for every $ k $-degenerate graph $ G $. %The arboricity $ a（G）$ of a graph $ G $ is the minimum number of forests into which its set of edges can be partitioned. %If $ G $ is $ k $-degenerate, the…
The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses
我们用$χ$ g（G）表示图G的游戏色数，它是Alice在G上赢得着色游戏所需的最小颜色数.我们从Montassier et al. [M. Montassier，P. Ossona de Mendez，A. Raspaud和X. Zhu，Decomposing a graph into forests，J. Graph Theory Ser. B，102（1）：38-52，2012]，并且独立地，从Wang和Zhang，[Y. Wang和Q.张某Decomposing a planar graph with girth at least 8 into a forest and a matching，Discrete Maths，311：844-849，2011] that planar graphs with girth at least 8 have game chromatic number at most 5.人们可能会问，对于足够大的围长，这个5的界是否可以改进。在本文中，我们证明它不能。不仅如此，我们还证明了存在仙人掌CT（即每条边最多只属于一个圈的图）的$χ$ g（CT）= 5，尽管其圈长任意大，甚至圈间距离任意大.
