Degrees are Useless in SNORT When Measuring Temperature

TL;DR

This paper investigates the temperature of positions in the game Snort, revealing that it can grow much faster than the degree of the graph, challenging previous assumptions about the relationship between degree and temperature.

Contribution

The paper constructs a family of Snort positions where the temperature exceeds the degree, demonstrating that degree is not a reliable measure of urgency in these cases.

Findings

Temperature can be infinitely larger than the degree in Snort positions.

Constructed examples show temperature grows twice as fast as the degree.

Challenges assumptions about degree as a measure of urgency in combinatorial game theory.

Abstract

Snort is a two-player game played on a simple graph in which players alternately colour a vertex such that they do not colour adjacent to their opponents' vertex. In combinatorial game theory, the temperature of a position is a measure of the urgency of moving first. It is known that the temperature of \snort in general is infinite ($K_{1,n}$ has temperature $n$). We show that the temperature in addition can be infinitely larger than the degree of the board being played on. We do so by constructing a family of positions in which the temperature grows twice as fast as the degree of the board.