The study of a variation of the marking game, in which the first player marks vertices and the second player marks edges of an undirected graph was proposed by Bartnicki et al. (Electron J Combin 15:R72, 2008). In this game, the goal of the second player is to mark as many edges around an unmarked vertex as possible, while the first player wants just the opposite. In this paper, we prove various bounds for the corresponding graph invariant, the vertex-edge coloring number \({\text {col}}_\mathrm{ve}(G)\) of a graph G. In particular, every (finite or infinite) graph G whose edges can be oriented in such a way that the maximum out-degree is bounded by an integer d has \({\text {col}}_\mathrm{ve}(G)\le d+2\). We investigate this invariant in (classes of) planar graphs, including some infinite lattices. We present a close connection between the vertex-edge coloring number of a graph G and the game coloring number of the subdivision graph S(G). In our main result, we bound the vertex-edge coloring number in complete graphs from below and from above, and while \({\text {col}}_\mathrm{ve}(K_n)\le \lceil \log _2{n}\rceil +2\), the difference between the upper and the lower bound is roughly \(\log _2(\log _2 n)\). The latter results are, in fact, true for any multigraph whose underlying graph is \(K_n\).

Bartnicki等人提出了一种关于标记游戏的变体的研究，其中第一个玩家标记顶点，第二个玩家标记无向图的边缘。（Electron J Combin 15：R72，2008）。在此游戏中，第二位玩家的目标是在未标记的顶点周围标记尽可能多的边缘，而第一位玩家则希望相反。在本文中，我们证明了对应图表不变各种界限，点-边着色数\（{\文本{COL}} _ \ mathrm {已经}（G）\）的曲线图的G ^。尤其是，每个（有限或无限）图G的边都可以以最大出度由整数d限制的方式定向，它们具有\（{\ text {col}} _ \ mathrm {ve}（G ）\ le d + 2 \）。我们在平面图（的类）中研究此不变性，包括一些无限晶格。我们提出了图G的顶点边缘着色数与细分图S（G）的游戏着色数之间的紧密联系。在我们的主要结果中，我们将顶点边缘着色数从下面和上面绑定到完整图形中，而\（{\ text {col}} _ \ mathrm {ve}（K_n）\ le \ lceil \ log _2 { n} \ rceil +2 \），上限和下限之差大致为\（\ log _2（\ log _2 n）\）。实际上，对于任何基础图为\（K_n \）的多重图，后一个结果都是正确的。