The irregularity strength of dense graphs -- on asymptotically optimal solutions of problems of Faudree, Jacobson, Kinch and Lehel
Abstract
The irregularity strength of a graph $G$, $s(G)$, is the least $k$ such that there exists a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ attributing distinct weighted degrees to all vertices, or equivalently the least $k$ enabling obtaining a multigraph with nonrecurring degrees by blowing each edge $e$ of $G$ to at most $k$ copies of $e$. In 1991 Faudree, Jacobson, Kinch and Lehel asked for the optimal lower bound for the minimum degree of a graph $G$ of order $n$ which implies that $s(G)\leq 3$. More generally, they also posed a similar question regarding the upper bound $s(G)\leq K$ for any given constant $K$. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order $\frac{1}{K-1}n$ for every fixed integer $K\geq 3$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.09584
- arXiv:
- arXiv:2406.09584
- Bibcode:
- 2024arXiv240609584P
- Keywords:
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- Mathematics - Combinatorics;
- 05C15;
- 05C78