Jacobsthal numbers in generalised Petersen graphs

TL;DR

This paper establishes a precise relationship between the number of 1-factorizations in certain generalized Petersen graphs and Jacobsthal numbers, and confirms the list coloring conjecture for these graphs.

Contribution

It proves a new exact formula linking 1-factorizations of GP(3k,k) graphs to Jacobsthal numbers and verifies the list coloring conjecture for these graphs.

Findings

Number of 1-factorizations equals J(k) for odd k

Number of 1-factorizations equals 4J(k) for even k

List coloring conjecture verified for GP(3k,k) graphs

Abstract

We prove that the number of $1$-factorisations of a generalised Petersen graph of the type $GP(3k,k)$ is equal to the $k$th Jacobsthal number $J(k)$ if $k$ is odd, and equal to $4J(k)$, when $k$ is even. Moreover, we verify the list colouring conjecture for $GP(3k,k)$.