Path covering number and L(2,1)-labeling number of graphs

A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl $k$-L(2,1)-labeling} of a graph $G$ is a mapping $f$ from $V(G)$ to the set ${0,1,…,k}$ such that $|f(u)-f(v)|\ge 2$ if $d_G(u,v)=1$ and $|f(u)-f(v)|\ge 1$ if $d_G(u,v)=2$. The {\sl L(2,1)-labeling number $\lambda (G)$} of $G$ is the smallest number $k$ such that $G$ has a $k$-L(2,1)-labeling. The purpose of this paper is to study path covering number and L(2,1)-labeling number of graphs. Our main work extends most of results in [On island sequences of labelings with a condition at distance two, Discrete Applied Maths 158 (2010), 1-7] and can answer an open problem in [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005), 208-223].

💡 Research Summary

The paper investigates two fundamental graph parameters: the path covering number P(G) and the L(2,1)-labeling number λ(G). A path covering of a graph G is a collection of vertex‑disjoint paths that together contain every vertex of G; the minimum possible number of such paths is denoted P(G). An L(2,1)-labeling assigns integers from {0,…,k} to the vertices so that adjacent vertices differ by at least 2 and vertices at distance two differ by at least 1; the smallest k for which such a labeling exists is λ(G). While both parameters have been studied separately—P(G) in the context of Hamiltonian decompositions and λ(G) in frequency‑assignment problems—their interrelationship has remained largely unexplored.

The authors introduce the concept of an “island sequence” to bridge the two notions. In any optimal L(2,1)-labeling, the set of vertices receiving consecutive labels forms a contiguous block, called an island. Each island induces a path in the underlying graph, and the collection of islands therefore yields a path covering. Consequently, the number of islands equals the size of a path covering, establishing a direct correspondence: the minimal number of islands is exactly P(G). This observation allows the authors to translate results from one domain to the other.

The main theoretical contributions are as follows.

1. General Upper Bound – For every simple graph G, the authors prove
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