On strong avoiding games

TL;DR

The paper analyzes strong Avoider-Avoider games on $E(K_n)$ and its CAvoider-CAavoider variants, focusing on two target properties $P_4$ and ${\cal CC}_{>3}$. It develops case-based, strategy-stealing arguments and structural lemmas from preliminaries to establish Blue as the winning player for several games: in the standard strong variant for the $P_4$ game ($n \ge 8$) and the ${\cal CC}_{>3}$ game ($n \ge 5$), and in the CAvoider-CAavoider setting for the $S_3$, $P_4$, and Cycle games. The results illuminate how half-move advantages and graph structure constraints drive outcomes and extend known results from Sim to broader graph avoidance games. These constructive strategies contribute to understanding win conditions and progression dynamics in strong positional games on graphs, with potential implications for related combinatorial game theory problems.

Abstract

Given an increasing graph property $\cal F$, the strong Avoider-Avoider $\cal F$ game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses $\cal F$ first loses the game. If the property $\cal F$ is "containing a fixed graph $H$", we refer to the game as the $H$ game. We prove that Blue has a winning strategy in two strong Avoider-Avoider games, $P_4$ game and ${\cal CC}_{>3}$ game, where ${\cal CC}_{>3}$ is the property of having at least one connected component on more than three vertices. We also study a variant, the strong CAvoider-CAvoider games, with additional requirement that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games $S_3$ and $P_4$, as well as in the $Cycle$ game, where the players aim at avoiding all cycles.

Figures (13)

• Figure 1: Case 1: (a) the graph before the second move of Blue. (b) The imagined graph before the second move of Red.
• Figure 2: Case 2: (a) the graph before the second move of Blue. (b) The imagined graph before the second move of Red.
• Figure 3: Case 3: (a) the graph before the second move of Blue. (b) The imagined graph before the second move of Red.
• Figure 4: Case 4: (a) the graph after the second move of Blue. (b) The possible moves of Blue if the rule 1 of Stage 1 is in order, shown as dashed lines.
• Figure 5: Case 1: (a) the graph before the second move of Blue. (b) The imagined graph before the second move of Red.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Claim 3.1
• proof
• Claim 4.1
• proof
• Claim 4.2