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

TL;DR

The paper studies extendibility of partial $\lambda$-labelings and $L(2,1)$-colorings by linking extendibility to Hamiltonian paths in the complement $\overline{G}$ and by developing a framework around line graphs of bipartite multigraphs, termed skew graphs, via a table representation. It proves that a partial labeling extends to a full labeling if and only if there exists a $\overline{G}$-Hamiltonian path consistent with the given distance pattern, connecting to a constrained disjoint-path formulation. It then analyzes skew graphs $\overline{L(H)}$ to obtain a practical, polynomial-time approach (time $O(pqn)$) to decide toughness, generate all Hamiltonian paths from a given vertex, and compute the path-covering number $\pi(G)=DF(G)+1$ and the maximum path length $\ell(G)=|G|-DF(G)-1$ via a deficit parameter $DF(G)$. These results yield concrete methods for computing $\lambda$-labelings and $L(2,1)$-colorings of line graphs of bipartite multigraphs and have applications to $K_n\Box K_n$ and related $\lambda$-rectangles, offering a robust combinatorial toolkit for these distance-constrained labeling problems.

Abstract

This paper deals with the $λ$-labeling and $L(2,1)$-coloring of simple graphs. A $λ$-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 $λ$-labeling $f$ is given in a graph $G$. A general question is whether $f$ can be extended to a $λ$-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 $λ$-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 $λ$-squares.

Figures (7)

• Figure 1: Does there exist a Hamiltonian path in the Petersen graph consistent with the above labels?
• Figure 2: An example of skew graph and its corresponding table
• Figure 3: $u, v, w, t$ are tough vertices but no Hamiltonian path starts from any of them
• Figure 4: From left to right the patterns $\boxplus$, ${\mathcal{R}}$, ${\mathcal{C}}$, ${\mathcal{R}}$$\cup$${\mathcal{C}}$
• Figure 5: The patterns involving in the proof of Proposition \ref{['square-case']}