# Counting independent sets and colorings on random regular bipartite graphs

## TL;DR

This paper develops efficient algorithms to approximately count independent sets and colorings on large random regular bipartite graphs, extending previous methods and confirming open questions in graph counting.

## Contribution

It introduces an FPTAS for counting independent sets and q-colorings on almost all large regular bipartite graphs, based on recent techniques.

## Key findings

- FPTAS for independent sets when Δ ≥ 53
- FPTAS for weighted independent sets with large Δ and λ
- FPTAS for q-colorings on large Δ-regular bipartite graphs

## Abstract

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every $\Delta$-regular bipartite graph if $\Delta\ge 53$. In the weighted case, for all sufficiently large integers $\Delta$ and weight parameters $\lambda=\tilde\Omega\left(\frac{1}{\Delta}\right)$, we also obtain an FPTAS on almost every $\Delta$-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all $q\ge 3$ and sufficiently large integers $\Delta=\Delta(q)$, there is an FPTAS to count the number of $q$-colorings on almost every $\Delta$-regular bipartite graph.