# $H$-colouring $P_t$-free graphs in subexponential time

## TL;DR

This paper demonstrates that the generating function for graph homomorphisms to certain multigraphs and the number of 3-colorings can be computed in subexponential time for $P_t$-free graphs, but 4-colorability remains computationally hard.

## Contribution

It introduces a subexponential time algorithm for counting homomorphisms and 3-colorings in $P_t$-free graphs, and establishes a linear bound on pathwidth in terms of maximum degree.

## Key findings

- Subexponential time algorithm for counting (list) homomorphisms to specific multigraphs.
- Subexponential time computation of the number of 3-colorings in $P_t$-free graphs.
- Hardness result: no subexponential algorithm for 4-colorability assuming ETH.

## Abstract

A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{O\left(\sqrt{tn\log(n)}\right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of $P_t$-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that $P_t$-free graphs have pathwidth that is linear in their maximum degree.