For edge-color-critical graphs, non-$r$-partite spectral extremal graphs are edge extremal

Abstract: A graph is non-$r$-partite if its chromatic number exceeds $r$. For an edge-color-critical graph $F$ with $χ(F)=r+1$, let $\mathrm{ex}<em>{r+1,ρ}(n,F)$ be the maximum adjacency spectral radius among non-$r$-partite $F$-free graphs of order $n$, and let $\mathrm{EX}</em>{r+1,ρ}(n,F)$ and $\mathrm{EX}<em>{r+1}(n,F)$ be the families of such graphs attaining, respectively, this maximum spectral radius and the maximum number of edges $\mathrm{ex}</em>{r+1}(n,F)$. Fang and Lin conjectured that $$\mathrm{EX}<em>{r+1,ρ}(n,F)\subseteq\mathrm{EX}</em>{r+1}(n,F)$$ for every such $F$ and all large $n$. In this paper, we prove this inclusion under the hypothesis $$\mathrm{ex}<em>{r+1}(n,F)=|E(T</em>{n,r})|-\lfloor n/r\rfloor+O(1)$$, where $T_{n,r}$ is the Turán graph, together with an embeddability condition on $F$. As the main application, for $F=K_{1,1,t_3,\ldots,t_{r+1}}$ with $t_3,\ldots,t_{r+1}\ge 2$ we show [ \mathrm{ex}{r+1}(n,F)=|E(T{n,r})|-\Bigl\lfloor\frac nr\Bigr\rfloor+2(t_{\min}-1), \qquad t_{\min}:=\min{t_3,\ldots,t_{r+1}}, ] for all sufficiently large $n$. We further identify the unique spectral extremal graph, so that in particular $$\mathrm{EX}<em>{r+1,ρ}(n,F)\subseteq\mathrm{EX}</em>{r+1}(n,F)$$.

• The paper establishes that spectral extremal graphs coincide with edge extremal graphs in non-r-partite settings for edge-color-critical forbidden subgraphs.
• It introduces a spectral reduction theorem that leverages s-embeddability and second-order Rayleigh analysis to translate spectral questions into edge-count problems.
• The study provides explicit constructions and unique characterizations for complete multipartite forbidden subgraphs, refining classical Turán results with exact edge count adjustments.

Non-$r$-Partite Spectral Extremal Graphs for Edge-Color-Critical Forbidden Subgraphs

Introduction and Context

This paper addresses the interaction between classical extremal graph theory and spectral graph theory in the non-$r$-partite setting, focusing on edge-color-critical forbidden subgraphs $F$ with chromatic number $\chi(F) = r+1$. Specifically, for graphs of large order $n$, the paper studies two extremal parameters in the class of non-$r$-partite, $F$-free graphs:

• The maximal number of edges ($\mathrm{ex}_{r+1}(n,F)$)
• The maximal adjacency spectral radius ($\mathrm{ex}_{r+1,\rho}(n,F)$)

The central conjecture (Fang-Lin) posits that for such $F$, the spectral extremal graphs (maximizing spectral radius) are contained in the edge extremal family (maximizing the number of edges) for sufficiently large $r$0; that is,

$r$1

The present work settles this conjecture for broad classes of edge-color-critical graphs under mild structural and extremal enumerative hypotheses.

Main Results

The principal technical advance is a spectral reduction theorem, formalized in Theorem 1, which removes the spectral question to the level of edge counting, contingent on:

• The edge extremal number for non-$r$2-partite, $r$3-free graphs is given by

$r$4

• $r$5 satisfies a combinatorial "embeddability" criterion (introduced as $r$6-embeddability), which encodes that the addition of a small "throttled" neighborhood to two parts of a near-Turán host does not introduce $r$7, but any further increase does.

The main application resolves the conjecture for complete multipartite graphs $r$8 with all $r$9, deriving the exact formula: $F$0 where $F$1. The paper further classifies the unique spectral extremal graph for these cases.

Techniques and Proof Structure

The arguments combine tools and perspectives from both edge extremal and spectral graph theory, leveraging and extending several classical results:

• Simonovits's Extension of Turán’s Theorem for edge-color-critical graphs: tells when the Turán graph is the unique edge extremal $F$2-free graph.
• Spectral Stability (Desai et al.): any graph maximizing spectral radius under a forbidden subgraph condition must be structurally close to Turán graphs.
• Second-order Rayleigh Principle: a residual version of the Rayleigh quotient (Temple-type inequality) to capture spectral gains in situations where first-order estimates are inconclusive (typically, in finely balanced structures).

A key conceptual contribution is the $F$3-embeddability notion, a structural property that governs the link between embeddability thresholds of the forbidden structure $F$4, the configurations that can escape $F$5, and the balancing of the near-Turán host.

The technical core of the paper is the reduction process that, starting from a spectral extremal graph, progressively sharpens the possible structure and proves that maximal spectral radius can only be achieved by graphs attaining the edge extremal number, provided the non-$F$6-partite penalty corresponds to throttling two different parts (rather than a single local edge deletion).

The analysis distinguishes two types of forbidden subgraphs:

• Size-type (multipartite): extremality is achieved by throttling two parts, and the spectral and edge extremal structures coincide.
• Edge-type (complete graphs $F$7, complete split graphs): extremality is achieved by deleting a single edge, for which previous work gives the spectral result, but which fall outside the precise mechanism of the present theorem.

For the multipartite case, the paper refines the local-to-global analysis and provides matching upper and lower bounds, relying on tight majorization arguments and greedy embedding/exclusion criteria.

Strong Claims and Contrasts

• The paper proves the spectral–edge extremal inclusion for a wide class of edge-color-critical graphs, validating the Fang-Lin conjecture in these cases.
• It establishes that for complete multipartite forbidden subgraphs with two singleton parts, the spectral extremal graph is unique (contrasting with the usual non-uniqueness in the edge extremal case).
• It highlights the necessity of both the "bounded penalty" condition and $F$8-embeddability: there are edge-color-critical graphs for which the penalty for being non-$F$9-partite is a single edge deletion (not two throttled parts), and the spectral reduction does not apply as-is.

Numerical Results and Explicit Graph Families

For $\chi(F) = r+1$0, the explicit extremal graphs are constructed by augmenting a Turán graph with a "throttled" configuration:

• A complete $\chi(F) = r+1$1-partite graph with balanced parts.
• One vertex attached to all vertices in $\chi(F) = r+1$2 parts; in each of the two remaining parts, attached to exactly $\chi(F) = r+1$3 vertices, where $\chi(F) = r+1$4.

The precise edge count is given by: $\chi(F) = r+1$5 for all sufficiently large $\chi(F) = r+1$6, and spectral maximization is achieved uniquely in this explicit configuration.

Implications and Prospects

Theoretical Significance

• The results confirm that, under combinatorially natural conditions, spectral extremality reduces to edge extremality in the non-$\chi(F) = r+1$7-partite case for critical forbidden subgraphs.
• The analysis sharpens the understanding of how spectral maximization interacts with local forbidden subgraph constraints, particularly in the 'tight penalty' regime where the defect from Turán's bound is additive.
• The structural characterization via $\chi(F) = r+1$8-embeddability provides a powerful framework to analyze and classify other families.

Practical and Methodological Impact

• The techniques of refined Rayleigh analysis and secular function comparison (bypassing inconclusive first-order perturbations) supply new tools for spectral extremal graph theory, especially for local modifications against global balancing.
• The results can be leveraged in algorithmic extremal problems in spectral graph theory for forbidden subgraph classes with similar local-to-global balancing effects.

Future Developments

• Extension of $\chi(F) = r+1$9-embeddability: Determining for which critical graphs this property holds, or generalizing the reduction to all edge-color-critical $n$0 where the non-$n$1-partite extremal number is additive in Turán's bound.
• Sharp bounds for other forbidden subgraphs: Particularly for forbidden graphs where the non-$n$2-partite penalty is larger or more complex, e.g., theta graphs.
• Spectral uniqueness: Examination of when the spectral extremal configuration is unique, even in families where the edge extremal set is not, and its implications for coding and stability problems.
• Applicability of second-order spectral techniques: Adoption and extension of the Temple-type residual arguments for further extremal-spectral problems where first-order estimates remain inconclusive.

Conclusion

The work rigorously establishes the correspondence between spectral and edge extremality in non-$n$3-partite graphs excluding edge-color-critical $n$4, given embeddability and edge-count conditions. By characterizing the structure of all extremal cases, advancing both spectral techniques and combinatorial majors, these results lay foundational ground for continued progress on extremal-spectral classifications of forbidden subgraph problems.