Heavy and Light Paths and Hamilton Cycles

TL;DR

This paper analyzes the expected weights of the heaviest and lightest paths of a given length in degree-3 graphs with exponential edge weights, with implications for Hamilton cycles and paths in random cubic graphs.

Contribution

It provides a simple proof relating path weights to the number of paths, with new insights into Hamilton paths and cycles in random cubic graphs and supercritical G(n,p).

Findings

Expected weight bounds for heaviest and lightest paths are established.

Results imply existence of Hamilton paths and cycles with certain weights in random cubic graphs.

Connections are made to the longest cycle problem in supercritical G(n,p).

Abstract

Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length $k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under the assumption that $f(G,u_0,k)=\omega(1)$) the expected weight of the heaviest path of length $k$ in $G$ starting from $u_0$ is at least \begin{align*} (1-o(1))\left(k+\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right), \end{align*} and the expected weight of the lightest path of length $k$ in $G$ starting from $u_0$ is at most \begin{align*} (1+o(1))\left(k-\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right). \end{align*} We demonstrate the immediate implication of this result for Hamilton paths and Hamilton cycles in random cubic graphs, where we show that typically there exist paths and cycles of such weight as well. Finally, we discuss the connection of this result to the question of a longest cycle in the giant component of supercritical $G(n,p)$.