Equalities Among Critical Biases in Positional Games

Definitions and mathematical statement

For a graph on a prescribed vertex set, let $H$ be the property of containing a Hamilton cycle, let $D_k$ be the property that every vertex has degree at least $k$, and let $C_k$ denote $k$-connectivity. Under the usual vertex-connectivity convention, $C_k$ means that the graph has more than $k$ vertices and remains connected after deletion of any set of fewer than $k$ vertices. The source does not specify whether this convention or edge connectivity is intended. Write $b_P$ for the critical bias of the positional game with target property $P$, using one fixed board and one fixed bias/turn convention for all compared properties. The two source questions are \[b_H=b_{D_2}\ ?\qquad\text{and}\qquad b_{C_k}=b_{D_k}\ ?\] These are written as equalities of critical biases, not merely equality of asymptotic leading terms. For example, in the customary $(1:b)$ Maker--Breaker edge game on $K_n$, Maker claims one free edge and Breaker claims $b$ free edges per round, and Maker seeks to achieve $P$ in Maker's own spanning graph; a common convention sets $b_P(n)$ to the largest integer $b$ for which Maker has a winning strategy, with zero if there is no positive winning bias. Under that convention the displayed questions compare $b_H(n)$ with $b_{D_2}(n)$ and $b_{C_k}(n)$ with $b_{D_k}(n)$. This is an explicit conditional illustration: the supplied extract does not identify the board, game type, first player, critical-bias convention, or quantifier range in $n$ and $k$, so it does not determine whether the intended equalities hold for every admissible size or only eventually. The supplied literature triage of 2026-08-21 labels the questions open and reports no definitive resolution located.

Short English statement

Ask whether the Hamiltonicity threshold bias equals the minimum-degree-two threshold, and whether the $k$-connectivity threshold equals the minimum-degree-$k$ threshold. The exact game conventions and parameter ranges are missing from the extract.

Sources

• Local UnsolvedMath registry, permanent ID 30000742, source problem catalogue entry, Equalities Among Critical Biases in Positional Games, proposed year 2007; no individual proposer supplied.
• Supplied citation: Mini-Workshop: Positional Games (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/20, pp. 1073--1114: https://doi.org/10.4171/owr/2007/20. Supplied literature label: original Oberwolfach report.