Minimum degree stability for graphs without odd-cycle blow-up

Abstract: For fixed integers $g\ge 2$ and $t\ge 1$, and every $\varepsilon&gt;0$, we prove that there exists a constant $ρ&gt;0$ such that every $n$-vertex graph $G$ with $δ(G)\ge (2/(2g+1)+\varepsilon)n$ either contains $C_{2g-1}[t]$, or can be made bipartite by deleting $O(n<sup>{2-ρ})$ edges. This gives an affirmative answer to a question of Illingworth in [Minimum degree stability of $H$-free graphs, Combinatorica, 43(1):129-147, 2023.]

• The paper establishes a polynomial upper bound on the number of edge deletions required to transform graphs with minimum degree above 2/(2g+1)+ε into bipartite structures when odd-cycle blow-ups are excluded.
• It leverages dense Boolean 2-CSP sampling techniques to transfer local bipartiteness deficits into global graph properties, facilitating precise extremal counts.
• Through careful reductions and comparative extremal bounds, the authors demonstrate the sharpness of the minimum degree threshold and outline prospects for refining the polynomial exponent.

Minimum Degree Stability for Graphs Without Odd-Cycle Blow-Up

Introduction and Context

This work addresses an extremal problem in the context of minimum degree stability for graphs excluding specific odd-cycle blow-ups. The classical Erdős-Stone-Simonovits theorem, which relates edge density with forbidden subgraphs determined by chromatic number, motivates the broader landscape of extremal graph theory. The study of stability, particularly with minimum degree conditions, characterizes how far nearly extremal graphs are from being $r$-partite or otherwise structured in presence of forbidden subgraphs.

A notable direction explored here concerns $C_{2g-1}[t]$-free graphs—graphs that avoid the $t$-blow-up of the odd cycle $C_{2g-1}$. Previously, Illingworth established that above the minimum degree threshold $2/(2g+1)$, such graphs can be made bipartite by deleting $o(n^2)$ edges, and posed whether a polynomial bound on the deletion distance is possible. This paper resolves that question affirmatively, providing a concrete polynomial bound on the number of edges required to render the graph bipartite.

Main Result

The principal theorem established is as follows: For every $g \ge 2$, $t \ge 1$, and $\varepsilon > 0$, there exist constants $C > 0$, $C_{2g-1}[t]$0 and $C_{2g-1}[t]$1 such that any $C_{2g-1}[t]$2-vertex graph $C_{2g-1}[t]$3 with minimum degree at least $C_{2g-1}[t]$4 either contains a copy of $C_{2g-1}[t]$5 or can be made bipartite by deleting at most $C_{2g-1}[t]$6 edges.

This result is tight with respect to the minimum degree threshold: the balanced blow-up of $C_{2g-1}[t]$7 matches the threshold asymptotically and cannot be made bipartite by deleting less than $C_{2g-1}[t]$8 edges, demonstrating optimality of the coefficient.

Technical Contributions

Sampling Argument for Bipartiteness Distance

The paper leverages dense Boolean $C_{2g-1}[t]$9-CSP sampling techniques, specifically invoking results from Alon-Fernandez de la Vega-Kannan-Karpinski, to show that the "distance from bipartiteness" ($t$0), is inherited at an appropriate scale for random induced subgraphs. Quantitatively, if a graph $t$1 has a certain distance from being bipartite ($t$2), then a random subset of $t$3 vertices will, with constant probability, witness a proportional bipartiteness deficit, facilitating local-to-global arguments.

Reduction and Application of H\"aggkvist's Theorem

A reduction to the two-connected non-bipartite case is developed, enabling application of H\"aggkvist’s theorem: sufficiently large 2-connected non-bipartite graphs above the minimum degree threshold must contain large odd cycles. Using a splitting process on components, the paper manages the combinatorial complexity introduced by non-2-connectedness and controls the size and degree inheritance properties required.

Quantitative Lower and Upper Bounds on Odd Cycles

A pivotal step is to show that, if a graph $t$4 above threshold and with large enough $t$5 does not contain $t$6, it must contain many copies of $t$7. Specifically, if the bipartiteness deficit $t$8 is large, $t$9 contains at least $C_{2g-1}$0 copies of $C_{2g-1}$1. This quantitative lower bound is then confronted with an upper bound for the number of $C_{2g-1}$2 copies in $C_{2g-1}$3-free graphs due to Alon and Shikhelman, which asserts a sub-power bound $C_{2g-1}$4. Comparing bounds yields the super-polynomial decay in $C_{2g-1}$5, closing the argument.

Quantitative and Structural Sharpness

The exponent $C_{2g-1}$6 for the deletion distance is explicitly bounded away from zero by a function of $C_{2g-1}$7, $C_{2g-1}$8, and $C_{2g-1}$9, and the threshold constant $2/(2g+1)$0 is demonstrated to be sharp via explicit extremal constructions. The paper also observes that while Allen's results for the chromatic threshold context admit sharp constants up to Zarankiewicz-type extremal functions, the full determination of the optimal polynomial exponent in this $2/(2g+1)$1-free setting remains open. The possibility of expressing it in terms of bipartite subgraph extremal functions or related Zarankiewicz numbers is suggested as a further direction for the field.

Implications and Future Directions

This result provides a polynomial bound on the bipartiteness deletion distance for graphs above the minimum degree threshold while excluding odd-cycle blow-ups. Practically, this strengthens local-to-global principles in extremal graph theory, offering constructive guarantees for graph decomposition and coloring under minimum degree constraints. Theoretically, the approach based on Boolean CSPs enriches the intersection between random sampling, property inheritance, and forbidden subgraph counts in dense settings.

Potential future work includes:

• Determining the precise exponent $2/(2g+1)$2 governing the polynomial bound and its relation to known extremal functions.
• Exploring whether analogous results are attainable for broader classes of forbidden subgraphs or under relaxed minimum degree hypotheses.
• Investigating algorithmic aspects related to efficiently finding a bipartition or the requisite set of edges to delete.

Conclusion

The paper resolves an open question on the polynomial minimum-degree stability for graphs excluding odd-cycle blow-ups. The methods combine delicate probabilistic sampling, deep structural reductions, and precise extremal counting, yielding the optimal threshold and a robust polynomial bound on the edge deletion distance to bipartiteness for this class of graphs. The results represent a substantive advance in stability theory, and the techniques developed are poised for further applications in fine-grained structural extremal combinatorics.