Games on deBruijn Graphs and Cycle Means

TL;DR

This paper introduces a novel game-theoretic method to assign edge weights in deBruijn graphs, ensuring uniform cycle averages, with explicit formulas and surprising linear relations, advancing graph weighting techniques.

Contribution

It presents a new approach using zero-sum games to determine edge weights in deBruijn graphs, ensuring equal cycle averages with explicit formulas.

Findings

Weights correspond to solving linear equations with fewer unknowns than cycles.

The method guarantees uniform average weights across all cycles.

Analysis of related graph games extends the theoretical framework.

Abstract

deBruijn graphs are widely used in genomics and computer science. In this paper we present a novel approach to finding weights on edges of doubly weighted deBruijn graphs. Given any fixed set of weights on vertices, we use a repeated two-person zero-sum game to find weights on edges so that every cycle on the deBruijn graph has the same average weight, providing explicit formulas. This approach uses minimax optimal strategies of the players. Once the weights on the edges are determined, we observe that they correspond to solving a set of linear equations with as many equations as there are cycles. This is very surprising, because there are many more cycles than unknowns. Moreover we analyze other, related games on graphs.