A Characterization of Triangle-Free Cyclic Graphs With Self-Loops Of Rank 3
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908556756779008 |
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| author | Lim, Johnny |
| author_facet | Lim, Johnny |
| contents | Let $G_S$ be a self-loop graph as the graph obtained by attaching a self-loop at every vertex in $S \subseteq V(G)$ of a simple graph $G.$ If $G=C_n$ is the cycle graphs of order $n$ and $S \neq \emptyset,$ we show that there are no rank 3 self-loop graphs $(C_n)_S$ for $n\geq 5.$ As a consequence, we determine and construct all possible rank 3 triangle-free self-loop cyclic graph of order at least 4 from $(C_4)_S$ via graph join operations. This provides a partial solution to the characterization problem of rank 3 self-loop graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Characterization of Triangle-Free Cyclic Graphs With Self-Loops Of Rank 3 Lim, Johnny Combinatorics 05C50, 05C90, 05C92 Let $G_S$ be a self-loop graph as the graph obtained by attaching a self-loop at every vertex in $S \subseteq V(G)$ of a simple graph $G.$ If $G=C_n$ is the cycle graphs of order $n$ and $S \neq \emptyset,$ we show that there are no rank 3 self-loop graphs $(C_n)_S$ for $n\geq 5.$ As a consequence, we determine and construct all possible rank 3 triangle-free self-loop cyclic graph of order at least 4 from $(C_4)_S$ via graph join operations. This provides a partial solution to the characterization problem of rank 3 self-loop graphs. |
| title | A Characterization of Triangle-Free Cyclic Graphs With Self-Loops Of Rank 3 |
| topic | Combinatorics 05C50, 05C90, 05C92 |
| url | https://arxiv.org/abs/2509.20158 |