Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916724467564544 |
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| author | Adak, Rajat Chandran, L. Sunil |
| author_facet | Adak, Rajat Chandran, L. Sunil |
| contents | For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erdős-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erdős-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles Adak, Rajat Chandran, L. Sunil Combinatorics Discrete Mathematics For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erdős-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erdős-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds. |
| title | Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2504.01501 |