On Hamiltonian-Connected and Mycielski graphs

TL;DR

This paper explores Hamiltonian-connected properties of graph squares, self-complementary graphs, and Mycielski graphs, proving new theorems that extend known results and confirming a conjecture about Hamiltonian-connectedness.

Contribution

It proves that squares of self-complementary graphs with more than 4 vertices are Hamiltonian-connected and confirms a conjecture relating Hamiltonian-connectedness of graphs and their Mycielski graphs.

Findings

Squares of self-complementary graphs (>4 vertices) are Hamiltonian-connected.

Mycielski graph of a k-critical graph is (k+1)-critical.

Confirmed the conjecture that the Mycielski graph of a Hamiltonian-connected graph (not K2) is Hamiltonian-connected.

Abstract

A graph $G$ is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of $G$. It is known that if $G$ is 2-connected then the graph $G^2$ is Hamiltonian-connected. In this paper we prove that the square of every self-complementary graph of order grater than 4 is Hamiltonian-connected. If $G$ is a $k$-critical graph, then we prove that the Mycielski graph $\mu(G)$ is $(k+1)$-critical graph. Jarnicki et al.[7] proved that for every Hamiltonian graph of odd order, the Mycielski graph $\mu(G)$ of $G$ is Hamiltonian-connected. They also pose a conjecture that if $G$ is Hamiltonian-connected and not $K_2$ then $\mu(G)$ is Hamiltonian-connected. In this paper we also prove this conjecture.