On a Vertex-Edge Marking Game on Graphs

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