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Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles

Rajat Adak, L. Sunil Chandran

TL;DR

This work introduces vertex-based localization to extend Erdős–Gallai bounds for paths and cycles. By defining p(v) as the maximum path length through v and c(v) as the maximum cycle length through v (or 2 if v is not in a cycle), the authors prove upper bounds |E(G)| ≤ ∑_{v} p(v)/2 and |E(G)| ≤ (∑_{v} c(v)/2) − c(u)/2, where c(u) is the graph circumference. They develop a comprehensive proof framework using longest v_0-paths, transforms, and a vertex-based counting scheme, and they prove that equality characterizes parent-dominated block graphs. The results recover the classical Erdős–Gallai bounds when p(v) and c(v) are globally bounded, and they establish a precise extremal-graph structure under the vertex-localization paradigm. Overall, the paper broadens localization from edges to vertices, connects local vertex parameters to global extremal bounds, and provides a robust extremal-graph classification for the vertex-based Erdős–Gallai inequalities.

Abstract

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.

Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles

TL;DR

This work introduces vertex-based localization to extend Erdős–Gallai bounds for paths and cycles. By defining p(v) as the maximum path length through v and c(v) as the maximum cycle length through v (or 2 if v is not in a cycle), the authors prove upper bounds |E(G)| ≤ ∑_{v} p(v)/2 and |E(G)| ≤ (∑_{v} c(v)/2) − c(u)/2, where c(u) is the graph circumference. They develop a comprehensive proof framework using longest v_0-paths, transforms, and a vertex-based counting scheme, and they prove that equality characterizes parent-dominated block graphs. The results recover the classical Erdős–Gallai bounds when p(v) and c(v) are globally bounded, and they establish a precise extremal-graph structure under the vertex-localization paradigm. Overall, the paper broadens localization from edges to vertices, connects local vertex parameters to global extremal bounds, and provides a robust extremal-graph classification for the vertex-based Erdős–Gallai inequalities.

Abstract

For a simple graph , let and denote the number of vertices and edges in , respectively. The Erdős-Gallai theorem for paths states that in a simple -free graph, , where denotes a path with length (that is, with edges). In this paper, we generalize this result as follows: For each , let be the length of the longest path that contains . We show that The Erdős-Gallai theorem for cycles states that in a simple graph with circumference (that is, the length of the longest cycle) at most , we have . We strengthen this result as follows: For each , let be the length of the longest cycle that contains , or if is not part of any cycle. We prove that where denotes the circumference of . \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

Paper Structure

This paper contains 10 sections, 22 theorems, 55 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Localization
  3. Our Result: Vertex-based localization
  4. Recovering the Classical Theorems
  5. Proof of \ref{['th:main']}
  6. Some Notations and Lemmas
  7. Some Useful Properties
  8. Proof of Inequality
  9. Characterizing the extremal graphs
  10. Proof of \ref{['th:main:path']}

Key Result

Theorem 1.1

(Turán turan1941egy) For a simple graph $G$ with $n$ vertices and clique number at most $r$, and equality holds if and only if $G$ is a regular Turán Graph with $n$ vertices and $r$ classes.

Figures (14)

  • Figure 1: Example of a block graph
  • Figure 2: Path $P'$ is a simple transform of $P$
  • Figure 4: If $|S|=1$, then $v$ is adjacent to all the vertices of $Back^*(P_v)\setminus \{v\}$
  • Figure 5: Boxed vertices represent the vertices in $S$. For $|S|>1$, $y$ is assumed to be the first vertex in $S\setminus \{w\}$ on $P_v$. With respect to $P_v$, $z_1 = Pred_y(1)$, $z_2 = Pred_y(2)$ and $v_1 = Pred_v(1)$.
  • Figure 6: Path $P_{z_2}$ can be constructed if $z_3$ exists in $Back^*(P_v)$ before $z_2$.
  • ...and 9 more figures

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 56 more