Graphs With Minimal Strength

TL;DR

This paper investigates the strength of graphs, providing conditions for minimal strength, solving existing questions, and establishing bounds, including for regular graphs and hypercubes, advancing understanding of graph labelings.

Contribution

It offers a sufficient condition for graphs to have strength equal to their order plus minimum degree, and characterizes graphs with minimal strength, solving prior open problems.

Findings

Established a condition for $str(G)=|V(G)|+\delta(G)$.

Proved every graph is either of this strength or a subgraph of such a graph.

Derived new lower bounds for the strength of various graphs, including hypercubes.

Abstract

For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in E(G)\},$ and the strength $str(G)$ of a graph $G$ itself is $str(G) = \min\{str_f(G)\;|\; f \mbox{ is a numbering of } G\}.$ A numbering $f$ is called a strength labeling of $G$ if $str_f(G)=str(G)$. In this paper, we obtained a sufficient condition for a graph to have $str(G)=|V(G)|+\d(G)$. Consequently, many questions raised in [Bounds for the strength of graphs, {\it Aust. J. Combin.} {\bf72(3)}, (2018) 492--508] and [On the strength of some trees, {\it AKCE Int. J. Graphs Comb.} (Online 2019) doi.org/10.1016/j.akcej.2019.06.002] are solved. Moreover, we showed that every graph $G$ either has $str(G)=|V(G)|+\d(G)$ or is a proper subgraph of a graph $H$ that has $str(H) = |V(H)| + \d(H)$ with $\d(H)=\d(G)$. Further, new good lower bounds of $str(G)$ are also obtained. Using these, we determined the strength of 2-regular graphs and obtained new lower bounds of $str(Q_n)$ for various $n$, where $Q_n$ is the $n$-regular hypercube.