# A Connected Version of the Graph Coloring Game

## TL;DR

This paper introduces a new variant of the graph coloring game where the colored vertices induce a connected subgraph, and studies its properties, establishing bounds for outerplanar and bipartite graphs.

## Contribution

It defines the connected game chromatic number and proves bounds for outerplanar graphs, also showing existence results for bipartite graphs with different winning strategies.

## Key findings

- Connected game chromatic number of outerplanar graphs is at most 5.
- Existence of outerplanar graphs with connected game chromatic number 4.
- Bipartite graphs exist where Bob wins with k colors, but Alice wins with 2 colors.

## Abstract

The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k $\ge$ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph.