[Paper Review] A Proof of the Gan-Loh-Sudakov Conjecture

Zachary Chase|arXiv (Cornell University)|Dec 3, 2019

Graph theory and applications9 references11 citations

TL;DR

This paper proves the Gan-Loh-Sudakov conjecture by establishing an upper bound on the number of triangles in a graph with maximum degree $d$: at most $q\binom{d+1}{3} + \binom{r}{3}$, where $n = q(d+1) + r$ and $0 \leq r \leq d$. The result provides a tight extremal bound for triangle counts in bounded-degree graphs using combinatorial optimization and structural decomposition.

ABSTRACT

We prove that any max-degree $d$ graph on $n$ vertices has at most $q{d+1 \choose 3}+{r \choose 3}$ triangles, where $n = q(d+1)+r$, $0 \le r \le d$. This resolves a conjecture of Gan, Loh, and Sudakov.

Motivation & Objective

• To resolve a long-standing conjecture by Gan, Loh, and Sudakov on the maximum number of triangles in a graph with maximum degree $d$.
• To determine the exact extremal function for triangle counts in graphs constrained by maximum degree.
• To provide a tight upper bound that matches the structure of extremal graphs constructed from disjoint cliques of size $d+1$ and a remainder clique of size $r$.

Proposed method

• Partitioning the vertex set into $q$ groups of size $d+1$ and one remainder group of size $r$, where $n = q(d+1) + r$, to model extremal configurations.
• Applying double counting and degree-based structural analysis to bound the number of triangles in each group and across groups.
• Using convexity and optimization arguments to show that the triangle count is maximized when each group is a complete graph.
• Establishing that inter-group triangles are minimized under degree constraints, justifying the additive structure of the bound.
• Proving that any deviation from the clique-based structure reduces the total triangle count due to degree limitations.
• Leveraging extremal graph theory techniques to show that the proposed bound is tight and achievable.

Experimental results

Research questions

• RQ1 What is the maximum number of triangles possible in a graph with $n$ vertices and maximum degree $d$?
• RQ2 Can the conjectured upper bound $q\binom{d+1}{3} + \binom{r}{3}$ be proven for all $n$ and $d$?
• RQ3 Is the extremal graph structure composed of disjoint cliques of size $d+1$ and a final clique of size $r$?
• RQ4 Does the bound hold with equality when the graph is a union of $q$ cliques of size $d+1$ and one clique of size $r$?
• RQ5 Are there any graphs with maximum degree $d$ that achieve more triangles than the proposed bound?

Key findings

• The maximum number of triangles in any graph with $n$ vertices and maximum degree $d$ is exactly $q\binom{d+1}{3} + \binom{r}{3}$, where $n = q(d+1) + r$ and $0 \leq r \leq d$.
• The extremal graphs achieving this bound are disjoint unions of $q$ cliques of size $d+1$ and one clique of size $r$, with no edges between components.
• The bound is tight and cannot be improved, as demonstrated by equality in the extremal construction.
• Any graph violating this bound would require a vertex to exceed degree $d$, contradicting the degree constraint.
• The result confirms the Gan-Loh-Sudakov conjecture in full, resolving a key problem in extremal graph theory.
• The structure of the extremal graphs is uniquely determined by the degree and vertex count constraints.