Random lifts of $K_5\setminus e$ are 3-colourable

TL;DR

This paper proves that the chromatic number of a random lift of the graph $K_5$ missing one edge is almost surely three as the lift size grows large.

Contribution

It provides a partial answer to whether random lifts of $K_5$ have a fixed chromatic number, showing that for $K_5$ minus an edge, it is almost surely three.

Findings

Random lifts of $K_5ackslash e$ are 3-colorable with high probability.

The result extends understanding of chromatic properties of random graph lifts.

Supports conjecture about fixed chromatic number in random lifts of complete graphs.

Abstract

Amit, Linial, and Matou\vsek (Random lifts of graphs III: independence and chromatic number, Random Struct. Algorithms, 2001) have raised the following question: Is the chromatic number of random $h$-lifts of $K_5$ asymptotically (for $h\to\infty$) almost surely equal to a single number? In this paper, we offer the following partial result: The chromatic number of a random lift of $K_5\setminus e$ is asymptotically almost surely three.