Connected Coloring Games on Corona Products of Graphs

Source abstract

We study the connected coloring game on corona products of graphs. For connected graphs $G$ and $H$, we prove the sharp upper bound $χ_{cg}(G\circ H)\leq\max\{Δ(G),Δ(H)+1\}+1$ and establish a further bound using the Bob-start ordinary coloring game on the second factor. We show that the difference between the Alice-start and Bob-start connected game chromatic numbers can be arbitrarily large in either direction. We further prove that a connected subgraph may have arbitrarily larger connected game chromatic number than its ambient connected graph, and that $χ_{cg}(G\circ H)$ cannot be bounded in terms of $χ_{cg}(G)$ and $χ_{cg}(H)$ alone, even when both factors have connected game chromatic number $2$. We also obtain corona products for which the difference $χ_g-χ_{cg}$ is arbitrarily large and construct infinitely many connected graphs with $χ_g<χ_{cg}$. Finally, for corona products of trees, we prove the universal upper bound $5$, characterize those attaining the value $3$, give structural sufficient conditions for the value $4$, and exhibit a broad family attaining the value $5$.