3.2 Mantel's Theorem and Triangle-Free Graphs

Triangle-free graphs are a fascinating subset of graph theory. They lack cycles of length 3, which impacts their structure and properties. Mantel's theorem sets an upper limit on the number of edges these graphs can have.

The theorem states that a triangle-free graph with n vertices can have at most ⌊n²/4⌋ edges. This limit shapes how these graphs look and behave, influencing their edge density, coloring, and overall structure.

Mantel's Theorem and Triangle-Free Graphs

Mantel's theorem and triangle-free graphs

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• Mantel's theorem establishes an upper bound on the number of edges in a triangle-free graph with $n$ vertices at most $\lfloor \frac{n^2}{4} \rfloor$ edges
• Triangle-free graphs contain no cycles of length 3 as subgraphs prohibiting the formation of triangular structures within the graph
• Maximum number of edges in a triangle-free graph directly proportional to the number of vertices in the graph constraining edge density based on vertex count
• Complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ serves as an example of a triangle-free graph achieving the maximum edge count specified by Mantel's theorem
  • Vertices divided into two disjoint sets with edges connecting vertices across sets but not within sets $($bipartite property$)$

Proof of Mantel's theorem

• Proof by contradiction assumes the existence of a triangle-free graph $G$ on $n$ vertices surpassing the edge limit of $\lfloor \frac{n^2}{4} \rfloor$
• Vertex $v$ in $G$ selected with maximum degree $\Delta$ representing the largest number of incident edges
• Edge count in $G$ bounded by $\Delta + \frac{(n-1-\Delta)(n-2)}{2}$ considering the remaining vertices forming a complete subgraph
• Exceeding $\lfloor \frac{n^2}{4} \rfloor$ edges requires $\Delta > \frac{n-1}{2}$ establishing a lower bound on the maximum degree
• Pigeonhole principle implies the existence of two adjacent neighbors of $v$ if $\Delta > \frac{n-1}{2}$ contradicting the triangle-free property
  • Pigeonhole principle states that placing $n$ items into $m$ containers with $n > m$ necessitates at least one container housing multiple items

Structure of triangle-free graphs

• Triangle-free graphs encompass both bipartite and non-bipartite graph structures
  • Bipartite graphs inherently triangle-free due to the absence of edges within vertex sets
  • Non-bipartite triangle-free graphs may contain odd cycles of length greater than 3 $($pentagons, heptagons, etc.$)$
• Girth of a triangle-free graph measures at least 4 representing the length of the shortest cycle present in the graph
• Chromatic number of a triangle-free graph limited to 2 colors sufficing for proper vertex coloring
  • Bipartite structure allows two-coloring with adjacent vertices assigned different colors
  • Non-bipartite triangle-free graphs convertible to bipartite by edge removal enabling two-coloring

Edge maximization in triangle-free graphs

• Mantel's theorem provides the formula for the maximum number of edges in a triangle-free graph with $n$ vertices $\lfloor \frac{n^2}{4} \rfloor$
• Problem-solving considerations for triangle-free graphs include:
  1. Parity of the vertex count $($odd or even$)$ affecting graph structure and edge distribution
  2. Graph classification as bipartite or non-bipartite influencing edge placement and cycle formation
  3. Interplay between vertex count, edge count, and vertex degrees shaping graph properties and constraints
• Example problem proves that a triangle-free graph with $n$ vertices and $\lfloor \frac{n^2}{4} \rfloor$ edges must exhibit a complete bipartite structure
  • Proof assumes graph $G$ satisfying the vertex and edge count conditions
  • Mantel's theorem confirms $G$ attains the maximum edge count for a triangle-free graph on $n$ vertices
  • Existence of non-adjacent vertices $u$ and $v$ allowing edge addition without triangle formation contradicts the maximum edge count assumption
  • Consequently, $G$ must adhere to a complete bipartite structure to maintain triangle-free property at maximum edge density