Erdős Conjecture and AR-Labeling
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912248229789696 |
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| author | Manattu, Arun J S, Aparna Lakshmanan |
| author_facet | Manattu, Arun J S, Aparna Lakshmanan |
| contents | Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an $AR$-labeling of $G$, if $f:E(G) \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an $AR$-vertex under $f$. The minimum $k$ such that there exists an $AR$-labeling $f:E\rightarrow \{1,2,3,\dots,k\}$ is called the $AR$-index of G, denoted by $ARI(G)$. In this paper, using a sequence originating from Erdős subset sum conjecture, a lower bound has been obtained for the $AR$-index of a graph and this bound is used to prove that only finitely many bistars, complete graphs and complete bipartite graphs are $AR$-graphs. The exact values of $AR$-index is obtained for stars and wheels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erdős Conjecture and AR-Labeling Manattu, Arun J S, Aparna Lakshmanan Combinatorics 05C78, 05C55 Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an $AR$-labeling of $G$, if $f:E(G) \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an $AR$-vertex under $f$. The minimum $k$ such that there exists an $AR$-labeling $f:E\rightarrow \{1,2,3,\dots,k\}$ is called the $AR$-index of G, denoted by $ARI(G)$. In this paper, using a sequence originating from Erdős subset sum conjecture, a lower bound has been obtained for the $AR$-index of a graph and this bound is used to prove that only finitely many bistars, complete graphs and complete bipartite graphs are $AR$-graphs. The exact values of $AR$-index is obtained for stars and wheels. |
| title | Erdős Conjecture and AR-Labeling |
| topic | Combinatorics 05C78, 05C55 |
| url | https://arxiv.org/abs/2502.19182 |