Andr{á}sfai--Erdős--Sós theorem under max-degree constraints

We establish the following strengthening of the celebrated Andr{á}sfai–Erdős–Sós theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $δ(G)$ and maximum degree $Δ(G)$ satisfy \begin{align*} δ(G) > \min \left{ \frac{3r-4}{3r-2}n-\frac{Δ(G)}{3r-2},~n-\frac{Δ(G)+1}{r-1} \right}, \end{align*} then $G$ is $r$-partite. This bound is tight for all feasible values of $Δ(G)$. We also obtain an analogous tight result for graphs with large odd girth. Our proof does not rely on the Andr{á}sfai–Erdős–Sós theorem itself, and therefore yields an alternative proof of this classical result.

💡 Research Summary

This paper presents a significant strengthening of the classical Andrásfai–Erdős–Sós theorem in extremal graph theory by incorporating the graph’s maximum degree into the condition that guarantees an r-partite structure.

The main result, Theorem 1.3, states that for an integer r ≥ 2, if G is an n-vertex K_{r+1}-free graph whose minimum degree δ(G) and maximum degree Δ(G) satisfy δ(G) > min{ (3r-4)/(3r-2) n - Δ(G)/(3r-2), n - (Δ(G)+1)/(r-1) }, then G is r-partite. The “min” function means the effective threshold for δ(G) is the larger (i.e., weaker) of the two bounds, which switch dominance depending on the value of Δ(G). The first term is active when Δ(G) is relatively small, resembling the original theorem but with an adjusted constant and a linear penalty in Δ(G). The second term becomes active for larger Δ(G), offering a more lenient condition on δ(G). The authors prove that this new bound is tight for all feasible values of Δ(G). They provide explicit, sophisticated constructions of extremal graphs using blowups of a specific graph W_r (the join of a 5-cycle and a complete graph K_{r-2}), with the part sizes meticulously calibrated as functions of Δ(G).

A parallel result, Theorem 1.4, is established for graphs with large odd girth. For k ≥ 2, if G is an n-vertex graph containing no odd cycles of length at most 2k+1 (C_{≤2k+1}-free) and satisfies δ(G) > min{ n/(k+1) - Δ(G)/(2k+2), n - (1+Δ(G))/k }, then G is bipartite. This result is also tight, with extremal examples based on blowups of an odd cycle C_{2k+3}, with constructions detailed separately for even and odd k.

A major contribution of this work lies in its proof methodology. The proofs of Theorems 1.3 and 1.4 are achieved via induction on r and k, respectively, and crucially do not rely on the original Andrásfai–Erdős–Sós theorems. Consequently, the paper provides novel, independent proofs of these classical results as a byproduct. The proof of Theorem 1.3 is particularly intricate. The base case (r=2, triangle-free graphs) is handled separately. In the inductive step, the graph G is assumed to be maximal K_{r+1}-free. The proof splits into two main cases corresponding to which term in the “min” is active. In each case, the strategy involves taking a vertex u of maximum degree Δ(G), analyzing the induced subgraph H on its neighborhood N(u), and using the inductive hypothesis to show H is (r-1)-partite. Leveraging the maximality of G, the existence of a K_{r-1} copy within N(u) is guaranteed. Then, using the pigeonhole principle and inclusion-exclusion arguments on the common neighborhoods of the vertices in this K_{r-1}, the remaining vertices are shown to form a single independent set, completing the r-partition. The second case involves additional complexity to handle a potential set of leftover vertices, leading to a final contradiction through a careful counting argument involving the sizes of specific common neighbor sets.

The proof of Theorem 1.4 also uses a case analysis based on whether a shortest odd cycle contains a vertex of maximum degree. It heavily utilizes a structural fact (Fact 3.1) about how a vertex can be adjacent to vertices on an odd cycle. By summing degrees over carefully selected vertices on a shortest odd cycle, the assumed conditions on δ(G) and Δ(G) are shown to lead to contradictions, forcing the conclusion that no odd cycle can exist, i.e., the graph is bipartite.

In summary, this research elegantly generalizes a foundational theorem by integrating the maximum degree parameter, yielding a precise, tight condition that reflects the nuanced interplay between the smallest and largest degrees in a graph. The work is complete with optimal constructions and self-contained proofs that offer fresh insight into the classical results it strengthens.