More relations between $\lambda$-labeling and Hamiltonian paths with emphasis on line graph of bipartite multigraphs

TL;DR

This paper explores the relationship between $\lambda$-labeling, Hamiltonian paths, and line graphs of bipartite multigraphs, providing algorithms and formulas for coloring problems and path coverings.

Contribution

It establishes a connection between extending partial $\lambda$-labelings and Hamiltonian paths, and offers polynomial algorithms for related graph coloring and path generation problems.

Findings

Extension of partial $\lambda$-labelings depends on Hamiltonian paths in complement graphs.

Derived formulas for path covering number and maximum path in complement graphs.

Developed polynomial algorithms for generating Hamiltonian paths and $\lambda$-squares.

Abstract

This paper deals with the $\lambda$-labeling and $L(2,1)$-coloring of simple graphs. A $\lambda$-labeling of a graph $G$ is any labeling of the vertices of $G$ with different labels such that any two adjacent vertices receive labels which differ at least two. Also an $L(2,1)$-coloring of $G$ is any labeling of the vertices of $G$ such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial $\lambda$-labeling $f$ is given in a graph $G$. A general question is whether $f$ can be extended to a $\lambda$-labeling of $G$. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of $G$. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in $L(2,1)$-coloring and $\lambda$-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph $K_n\Box K_n$ and the generation of $\lambda$-squares.