Strong Subgraph-Count Stability in $C_{2\ell+1}$-Free Graphs

Abstract: Starting from the stability theorem of Erdős and Simonovits, stability problems for graphs forbidding a fixed subgraph have been studied in terms of edge numbers, spectral radii and subgraph counts. Let $\mathcal{N}(F,G)$ denote the number of unlabeled copies of $F$ in $G$. It is known that, for every fixed path $P_t$ and even cycle $C_{2a}$, the maximum number of copies in an $n$-vertex $C_{2\ell+1}$-free graph is attained by the bipartite Turán graph $T_{n,2}$. In this paper we obtain strong structural stability for $C_{2\ell+1}$-free graphs in terms of copies of paths and even cycles. For fixed $\ell\ge2$ and $3\le r\le2\ell-1$, we show that if an $n$-vertex $C_{2\ell+1}$-free graph contains at least as many copies of $P_t$ or $C_{2a}$ as the corresponding suspended extremal construction, then it has the corresponding suspension structure. This gives exact high-chromatic extremal theorems for paths and even cycles. We also prove a counting theorem for nearly complete bipartite graphs. It shows that, for every fixed matching-admissible connected bipartite graph $F$, both imbalance between the two parts and missing cross-edges decrease the number of copies of $F$ by a term with a specified main coefficient. This theorem is independent of the forbidden odd cycle and converts subgraph-count assumptions into the edge bounds needed for the structural theorem.

• The paper establishes strong structural stability by proving that exceeding specific subgraph-count thresholds forces a precise combinatorial graph construction.
• It demonstrates that extremal graphs are characterized by a bipartite Turán core with suspended complete cliques, influenced by parameters such as r, t, and a.
• Refined counting lemmas yield sharp asymptotic formulas, connecting local subgraph-enumeration corrections to the global structure of C_{2l+1}-free graphs.

Strong Subgraph-Count Stability in $C_{2\ell+1}$-Free Graphs

Introduction and Context

This paper establishes precise structural stability results for $C_{2\ell+1}$-free graphs from the perspective of extremal subgraph counts. Building on the classical Erdős–Simonovits stability framework, the authors address the generalized Turán problem for the maximal number of copies of fixed paths ($P_t$) and even cycles ($C_{2a}$) in $C_{2\ell+1}$-free graphs, with a particular focus on high-chromatic settings and subgraph enumeration.

Previous works determined that the bipartite Turán graph $T_{n,2}$ is extremal for the maximum number of copies of $P_t$ or $C_{2a}$ in $C_{2\ell+1}$-free graphs for sufficiently large $n$ [Alon & Shikhelman, Gerbner, Hei & Hou, F\"uredi & Gunderson]. Further, recent progress leveraged subgraph-counting to infer graph structure in nearly extremal cases. Building on these, the present work advances both exact extremal enumerations and structural descriptions under strong stability conditions, especially for high-chromatic graphs and in terms of precise subgraph count thresholds.

Main Contributions

Structural Stability for Subgraph Counts

The paper proves that for fixed $C_{2\ell+1}$0 and $C_{2\ell+1}$1:

• Any $C_{2\ell+1}$2-vertex $C_{2\ell+1}$3-free graph $C_{2\ell+1}$4 with at least as many copies of $C_{2\ell+1}$5 or $C_{2\ell+1}$6 as the maximal possible in the corresponding extremal construction must itself belong to a specific suspension family characterized by a bipartite core with "suspended" complete graphs ("outside vertices").
• When considering high-chromatic graphs ($C_{2\ell+1}$7), the unique extremal graphs have a bipartite core $C_{2\ell+1}$8 (i.e., the Turán graph minus $C_{2\ell+1}$9 vertices), with a $P_t$0 clique suspended at a vertex, covering all $P_t$1 outside vertices.

The results yield exact stability: not only are extremal graphs determined up to $P_t$2 modifications, but their precise combinatorial construction is forced once the subgraph-count threshold is met.

Key Theorems

Paths

For fixed $P_t$3, $P_t$4 (with an additional technical restriction on $P_t$5 for odd $P_t$6), any $P_t$7-vertex $P_t$8-free graph with at least $P_t$9 copies of $C_{2a}$0 is either in the family $C_{2a}$1 or is one of a small collection ($C_{2a}$2) of extremal suspension graphs. The sharp dependence of the extremal family on $C_{2a}$3 and $C_{2a}$4 is clarified, including the effect of the parity of $C_{2a}$5 (even/odd).

Even Cycles

A similar theorem holds for even cycles $C_{2a}$6: if the subgraph count has reached the explicit threshold $C_{2a}$7, the structure of the graph must reflect the extremal bipartite core (with suspended components), again reducing to $C_{2a}$8 for $C_{2a}$9 and to a different family when $C_{2\ell+1}$0.

Counting Theorem for Near-Bipartite Graphs

A technical cornerstone is the local counting lemma: for any connected, matching-admissible bipartite $C_{2\ell+1}$1 and nearly complete bipartite host graphs $C_{2\ell+1}$2, the number of $C_{2\ell+1}$3-copies in $C_{2\ell+1}$4 can be tightly estimated as the corresponding value for $C_{2\ell+1}$5, minus explicit correction terms for both part-size imbalance and missing cross-edges, with leading constants determined by $C_{2\ell+1}$6. This enables conversion from subgraph-count assumptions to edge-count bounds necessary for stability arguments.

Asymptotic and Exact Formulas

Explicit asymptotic differences in the counted subgraphs between extremal and near-extremal graphs are provided. For example, for even paths,

$C_{2\ell+1}$7

and analogously for odd paths and even cycles, with full details on the coefficients. These distinguish among candidate extremal graphs when $C_{2\ell+1}$8 is large, but not infinite.

Numerical Strength and Exactness

The theorems identify, with tight asymptotics, when subgraph-count thresholds force not only the correct edge density, but full structural exactness up to the combinatorial type described (i.e., up to explicit suspension constructions).

A strong, sometimes counterintuitive, claim made is that for any fixed $C_{2\ell+1}$9 or $T_{n,2}$0, and for high chromatic number ($T_{n,2}$1), only the specific extremal suspension graphs can maximize subgraph counts, ruling out all others. The counting lemmas provide both main term and leading correction with precise constants, not just up to lower-order terms.

Implications and Future Directions

Practical/Combinatorial Implications

• For extremal graph theory, the results provide a blueprint for sharp "subgraph-count stability" in broad families of Turán-type problems. In practice, this allows classification of extremal and near-extremal $T_{n,2}$2-free graphs not just by edge density, but via richer subcombinatorial structure.
• This directly informs algorithms in extremal enumeration and in the design of graphs for purposes where controlling the count of certain motifs is critical (relevant, e.g., in network science, pseudorandomness, or design theory).

Theoretical Consequences

• These theorems extend and sharpen earlier generalized Turán-type results, extracting structural information from mere subgraph-count bounds.
• The fine-grained bounding method, in particular the reduction from subgraph-counts to edge-corrections via the matching-admissible graph lemma, is likely to be extensible to other forbidden structures and could inspire further generalizations.
• The paper leaves open (Problem 1) exact structural determination in the odd-path case for certain regimes, highlighting subtle dependencies between parity and coloring constraints.

Future Developments

• Potential extensions include similar stability phenomena for other families of forbidden subgraphs (e.g., more complex graphs beyond cycles and cliques), or to host graphs with more complex forbidden induced subgraphs.
• Further precision may be achievable for the unresolved cases, especially for odd paths with larger chromatic constraint, possibly via enhanced combinatorial or spectral techniques.
• The methods structurally connect extremal graph theory with spectral and counting bounds—this mixed toolkit may seed new results in related combinatorial optimization and random graph studies.

Conclusion

The paper achieves strong, technically detailed subgraph-count stability theorems for $T_{n,2}$3-free graphs, and gives sharp structural and enumerative characterization for extremal graphs in terms of path and cycle subgraph counts. The key conceptual advance is in relating local subgraph enumeration to global structure under forbidden subgraph constraints, opening avenues both for further classification results in extremal graph theory and for practical enumeration problems in large networked systems.

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Reference: "Strong Subgraph-Count Stability in $T_{n,2}$4-Free Graphs" (2607.04347)