Papers

A graph $G$ is called $CIS$ if each maximal clique intersects each maximal stable set of $G$, with maximality taken with respect to set inclusion. CIS graphs resemble perfect graphs in several respects and have interesting applications in game theory. The complexity of recognizing CIS graphs was posed as an open problem by Chvátal in the 1990s and has since led to conflicting conjectures. We settle the problem by showing that recognizing CIS graphs is $\mathsf{coNP}\text{-complete}$.

We introduce the notion of "$k$-thresholds” and show that Talagrand's discrete convexity conjecture is equivalent to the assertion that, for some universal integer $k \ge 2$, the $k$-threshold of every increasing family is at most a universal constant times its expectation threshold. We prove a reduction theorem that bounds the $k$-threshold of any increasing graph property in terms of ordinary thresholds of graphs in suitable decompositions of its members. As a consequence, we determine, up to a constant factor, the $k$-threshold of every fixed graph in terms of a natural $k$-density parameter. We also prove that $k=2$ suffices for several classical spanning graph containment properties. More generally, we establish the conjectured comparison between $k$-thresholds and expectation thresholds for broad classes of graph containment properties whose target graphs have low degeneracy.

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian $\Delta^H_k$ of a finite simplicial complex $\Sigma.$ As a consequence, we obtain new vanishing criteria for cohomology groups $H^k(\Sigma,\mathbb{R)}$ and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.

We introduce and start the study of a variant of packing functions in graphs. Given a graph $G$ with vertex set $V$ and nonnegative integer vectors $\mathbf{k}=(k_v)_{v\in V}$, $\boldsymbol\ell=(l_v)_{v\in V}$ and $\mathbf{u}=(u_v)_{v\in V}$, a function $f : V \rightarrow \mathbb{Z}_0^+$ is a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of $G$ if $l_v\leq f(v)\leq u_v$ for every $v\in V$ and the sum of the values of $f$ over the closed neighborhood of vertices $v$ with $f(v) = u_v$ is at most $k_v$. The weight of $f$ is the value $f(V) = \sum_{v\in V} f(v)$. We study the associated decision problem (RPP), which asks, given $G$, $\mathbf{k}$, $\boldsymbol\ell$, $\mathbf{u}$ and an integer number $x$, whether $G$ admits a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of weight at least $x$. We relate RPP to the $r$-dependent set problem, derive several NP-hardness results, model RPP as a compact (polynomial in size) Integer Linear Program, and take the first steps of a polyhedral study.

For $s\ge1$, let $G_s$ be the bipartite graph between the $s$-subsets and the $(s-1)$-subsets of $[2s]$, where adjacency means disjointness, and let $w(s)$ be the maximum number of $s$-subsets on an induced path in $G_s$. We prove $w(s)\ge \frac{4^s}{2048s^{5/2}}$ for all $s\geq 6$. This implies $\sup_{s\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $\Omega(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \[ \max\{\cc(\overline{P_n}),\ \cc(\overline{C_n})\} \le \log_2 n+\frac52\log_2\log_2 n+O(1). \] Together with the known lower bounds, this settles a conjecture of de Caen, Gregory, and Pullman (1985) and gives \[ \cc(\overline{P_n})=\log_2 n+\Theta(\log_2\log_2 n), \qquad \cc(\overline{C_n})=\log_2 n+\Theta(\log_2\log_2 n). \] We also give an independent proof of the latter order estimates. It uses a Hamiltonicity result of Kneser graphs and a key lemma proved by the Lovász local lemma.

Derangements are permutations without fixed points, and are in bijection with desarrangements: permutations whose first non-descent is even, or equivalently, permutations without “pixed points”. Bsila, Cox, Hugo, Styron, and Zhuang recently proved a theorem characterizing all $\Pi\subseteq\mathfrak{S}_{3}$, such that $1\leq\left|\Pi\right|\leq3$, for which the number of derangements avoiding all patterns in $\Pi$ is equal to the number of desarrangements avoiding all patterns in $\Pi$. They left finding a bijective proof of this theorem as an open problem, and posed a related conjecture concerning the distributions of fixed points and pixed points over pattern avoidance classes. In this paper, we give bijective proofs of this theorem and conjecture.

We consider a bounded set $P \subset \mathbb{R}^d$ and the lattice-point enumerator $L_P(t) = |tP \cap \mathbb{Z}^d|$ for real $t > 0$. We show that if two bounded measurable sets with boundary of measure zero have the same real-parameter lattice-point enumerators for all integer translates, then their indicator functions agree almost everywhere. As a corollary, any convex body is uniquely determined by this data. Our proof is short and Fourier-analytic, where the key device is a periodic point-counting function whose Fourier coefficients recover the Fourier transform of the indicator function on a dense set. This recovers and extends, with a unified argument, the uniqueness results for rational polytopes and symmetric convex bodies established by Royer [arXiv:1712.01973, arXiv:1712.03937], whose proofs relied on intricate case-specific geometric constructions.

This paper presents a classification of induced centrosymmetric alternant codes through the study of the automorphism structures inherited from Generalized Reed–Solomon (GRS) codes. We introduce the twisted conjugation action naturally associated with projective semilinear transformations and establish its correspondence with ordinary conjugacy in the projective semilinear group. This correspondence enables the application of Shintani's theorem to classify the $\gamma_{p^j}$-similarity classes of involutions. As a consequence, we obtain necessary and sufficient conditions for an alternant code to admit a centrosymmetric structure induced by a projective semilinear automorphism. The resulting classification unifies the different families of induced centrosymmetric alternant codes within a common automorphism-based framework.

We study the graphic $s$-$t$ path TSP on subcubic graphs (maximum degree 3): given distinct vertices $s,t$, find a shortest $s$-$t$ walk that visits every vertex. We prove an upper bound with the asymptotically optimal leading coefficient $5/4$ for every terminal pair, even when $G-\{s,t\}$ is disconnected. Specifically, every simple 2-connected subcubic graph $G$ on $n$ vertices has a spanning $s$-$t$ walk of length at most $\lfloor(5n+n_2(G))/4\rfloor$, where $n_2(G)$ counts its degree-2 vertices. An $O(n^2)$-time algorithm attains this bound. Combining an edge-rooted even-cover theorem of Wigal, Yoo, and Yu (WYY) with an even-cover-to-walk lemma proved here yields this bound for adjacent terminals, a consequence not stated explicitly in their paper. We extend the bound to arbitrary terminal pairs. For cubic graphs, it becomes $\lfloor 5n/4 \rfloor$, to our knowledge the first direct $5/4$ bound for cubic path TSP that does not use the general path-to-tour reduction.

In the unbiased Chooser-Picker (also known as Client-Waiter) game played on the edge set of a graph, Picker offers a pair of unclaimed edges in each turn, Chooser claims one, and the remaining edge goes back to Picker. We study the Chooser-Picker (C-P) degree game played on $d$-regular graphs, where Chooser aims to maximize the maximum degree of their induced subgraph, and Picker's objective is to defend every vertex by securing a certain minimum degree in Picker's own subgraph. While classical static pairing strategies guarantee a minimum degree of at least $\lfloor d/4 \rfloor$ for Breaker on general $d$-regular graphs in Maker-Breaker (M-B) games and for Picker in C-P games, outperforming this threshold has been a major open challenge in both frameworks. According to the foundational monograph of J. Beck, this challenge stands as the first among the seven most humiliating problems in combinatorial game theory. Our main result is that Picker can beat the $d/4$ bound. First, we prove that Picker can always guarantee a degree of at least one at every vertex on any $3$-regular graph. Based upon this we introduce a direct strategy to prove that Picker can secure a degree of at least $\lfloor d/3 \rfloor$ at every vertex for any $d$-regular graph. This highlights a fundamental structural advantage that Picker usually possesses over Breaker in sparse local games.

The study of coordinator polynomials of Weyl group lattices was initiated by Conway and Sloane. In 2013, by using a trigonometric substitution approach, Wang and Zhao proved the real-rootedness of the type $D$ coordinator polynomials. Subsequently, Xie and Zhang proved the compatibility between the types $C_n$ and $D_n$ coordinator polynomials. In this paper, we discover the interlacing pattern between the types $C_n$ and $D_n$ coordinator polynomials, which may be named as the second pattern of interlacing zeros.

We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima–Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.

For a given $d$-regular graph $G$, a Maker-Breaker degree game is played by two players who alternately claim previously unclaimed edges of $G$. In the standard variant, the goal of Maker is to maximize the maximum degree of their induced subgraph, while Breaker aims to minimize it, or equivalently, to guarantee a certain minimum degree in their own subgraph. A classic pairing strategy shows that Breaker can secure at least $\lfloor d/4 \rfloor$ edges at every vertex of any $d$-regular graph. Breaking this bound for general or even for specific classes of graphs has been a long-standing open problem in combinatorial game theory; indeed, J. Beck characterized this challenge in his monograph as the first among the seven most humiliating open problems of positional game theory. In this paper, we improve the $d/4$ bound for some infinite graph families, such as the hypercube graph $Q_d$, grids and tori. We first show that Breaker can secure a degree of one at every vertex in $Q_3$, then lift this to higher dimensions, where Breaker can guarantee a degree of at least $\lfloor d/3 \rfloor$.

A classical problem in lattice path enumeration counts paths that remain on one side of a boundary line. We study several classes of paths where this boundary is porous and show that they are related through a single half-turn rotation bijection. As a first application, we enumerate minimal bridges by relating them to excursions: for positive integers $k$ and $n$, the number of paths from $(0,0)$ to $(kn,n)$ with unit right and up steps that avoid all other lattice points on the line $y=x/k$ is $\frac{k}{kn+n-1}\binom{kn+n}{n}$. The same bijection yields a relation between the ordinary generating functions for binomial coefficients and $k$-Catalan numbers through a dual edge-forbidden model, extends to forbidden strips containing the diagonal, and handles a rational-slope case involving Duchon paths. Finally, our bijection also proves that the number of bridges from $(0,0)$ to $(2n,2n)$ that avoid even diagonal points is $C_{2n}+4C_{2n-1}$, with $C_n$ the $n$th Catalan number. This complements a result of Shapiro.

Let $B_{n,k}$ be the graph obtained from the balanced complete bipartite graph on $n$ vertices by deleting a matching of size $k$. If $G$ is an $n$-vertex triangle-free graph with $\kappa(\comp G)\geq k$, we prove that $\rhoA(G)\leq\rhoA(B_{n,k})$ for $n\geq4k+2$, with equality precisely when $G\cong B_{n,k}$, and we compute $\rhoA(B_{n,k})$ explicitly. We also solve the bipartite problem for every $n\geq2k+1$, determine the boundary value $\operatorname{spex}_{\kappa}(2k,K_3;k)=k-1$, and settle the full problem for $k=2$. In particular, $B_{n,2}$ is uniquely extremal exactly from order $6$ onward. For $k=1$, equivalently when the complement is connected, $B_{n,1}=K_{\ceil{n/2},\floor{n/2}}-e$ is uniquely extremal for every $n\geq3$.

A proper conflict-free (PCF) $k$-coloring of a graph $G$ is a proper $k$-coloring such that there exists a color that appears exactly once in the neighborhood of every non-isolated vertex $v\in V(G)$. The PCF chromatic number, denoted by $\chi_{pcf}(G)$, is the least integer $k$ such that there exists a PCF $k$-coloring of $G$. Given a graph $G$ and a positive integer $k$, PCF $k$-COLORABILITY is to decide whether $G$ admits a PCF $k$-coloring. Ahn et al. [Discrete Appl. Math. 377 (2025) 10-17] proved that PCF $k$-COLORABILITY is NP-complete for bipartite graphs. We strengthen this result by proving that PCF $k$-COLORABILITY is NP-complete for perfect elimination bipartite graphs, which is a proper subclass of bipartite graphs. We also show that the PCF chromatic number of a graph cannot be approximated within $O(n^{1-\varepsilon})$ unless P=NP, for any $\varepsilon>0$. On the positive side, we provide linear-time algorithms for PCF $k$-COLORABILITY in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs. We show that $\chi_{pcf}(G)\leq \omega(G)+1$ for block graphs, proper interval graphs, and pseudo-split graphs (except $C_5$), and we characterize all graphs for which the equality holds.

Let P be a finite spherical set. We prove that if P admits a solvable group of isometries acting transitively on it, then every r-coloring of a sufficiently high-dimensional sphere of radius slightly larger than the circumradius of P contains a monochromatic congruent copy of P. Our proof builds on the group-theoretic argument of Kriz and combines it with a topological method that may be of independent interest.

In recent years, integrable hierarchies have been used to great advantage for the enumeration of combinatorial maps. They have led to recurrence formulas with respect to the size and genus of the maps, e.g. for triangulations, bipartite quadrangulations and bipartite maps, and for constellations. These formulas are not only remarkably simple but also provide the fastest way of calculating these numbers of maps. With the exception of Louf's work on constellations, it has however remained a challenge to obtain recurrence formulas that control the degrees of the faces of the maps. Here we show how to achieve this for bipartite maps with bounded face degrees. By combining equations from the KP hierarchy and from the Virasoro constraints, a differentially algebraic system is obtained. It couples the generating functions of bipartite maps with bounded root face degrees while controlling the numbers of edges, black vertices, white vertices and number of faces of each degree (and in particular the genus). Finally, this system of ODEs is shown to give recurrence formulas that allows to calculate all the corresponding numbers of maps.

For an integer $k\geq2$, let $\chi_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897–904), which states that $\chi_k'(G)\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\{k,2k\}$ and a specified vertex partition. With a suitable choice of the part sizes, if the number of edges inside one part is $o(k^2)$, then $\chi_k'(G)\geq(4-2\sqrt2+o(1))k$. This gives connected bipartite and connected nonbipartite counterexamples. In particular, the same examples also disprove the earlier conjecture of Botler, Colucci, and Kohayakawa (The mod $k$ chromatic index of graphs is $O(k)$, J. Graph Theory 102 (2023) 197–200), which states that $\chi_k'(G)\leq k+C$ for some absolute constant $C$.

We introduce zero-sum inverse realization of property (P) of a graph $G$, obtained by imposing an additional condition \( \mathbf{1}^{\top}A^{-1}\mathbf{1}=0, \) on a matrix $A\in S(G)$ realizing property (P), where $\mathbf{1}$ is the all-ones vector. We prove that every graph of order at least three having property (P) admits such a zero-sum inverse realization. As applications, we prove that property~(P) is preserved under the join of two graphs of order at least $3$ and, more generally, under the $H$-join of a family of graphs of order at least $3$, where $H$ is arbitrary. Consequently, we obtain sufficient conditions for cographs and lexicographic products of graphs to possess property~(P). Throughout the paper, many examples are given.

Let $M_n$ be the minimum spanning tree of the complete graph $K_n$ with i.i.d.\ uniform edge weights. For a fixed forest $F$ with connected components $T_1, \ldots, T_d$, we show that there exists a function $\Psi$ on finite trees such that $$ n^{|E(F)|} \mathbb{P}_n(F \subseteq M_n) \longrightarrow \prod_{i=1}^d \Psi(T_i). $$ We give a recursive description of $\Psi$ and calculate it explicitly for several small trees. For the star $S_k$ and the path $P_k$, we prove that $\Psi(S_k) \sim \zeta(2)^k$ and $\Psi(P_k) \sim k^2/12$, respectively. We also show that the expected size of a ball of radius $r$ is asymptotic to $r^3/36$, and give exponential tail bounds.

In this paper, we investigate zero transfer on mixed graphs. Zero transfer is a quantum walk phenomenon in which the transition amplitude between two vertices is identically zero for all times, so that no quantum state transfer occurs between them. Using the Hermitian adjacency matrix, we derive necessary and sufficient conditions for zero transfer in mixed graphs. We then specialize these criteria to oriented circulant graphs, obtaining nonexistence results for prime order, structural restrictions for even order, and exhaustive computational classifications for small orders.

The smallest positive eigenvalue $\partial_2$ of the normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}$ of a connected graph is called its spectral gap and is intimately related to the Cheeger constant of $\mathcal{D}^{\mathcal{L}}$. Byrne, Johnston, Schildkraut and Tait (2025) conjectured that \[ \partial_2 \ge \frac{2}{3}\] for all connected graphs. We prove the following stronger result: for any connected graph $G$ of order at least 2, \[\partial_2 \ge \frac{2}{3} + \frac{4}{3\,t_{\max}},\] where $t_{\max}$ denotes the maximum transmission in $G$. Moreover, equality holds if and only if $G\cong K_{m,m}$ for some $m\ge 1$.

A word over an alphabet $\mathbb A$ contains a square if it has a subword of the form $XX$ where $X$ is a word. A word $W$ is extremal square-free if it does not contain a square, but it contains a square as soon as any letter of $\mathbb A$ is inserted at any position of $W$. Grytczuk, Kordulewski, and Niewiadomski conjectured that there are no extremal square-free words over alphabets of size at least 4. We prove this for alphabets of size at least 5. Our proof also implies that the sequence of nonchalant words defined by Grytczuk, Kordulewski, and Niewiadomski is infinite and converges to an infinite word for all alphabets of size at least 5.

We identify ordinary skew Schur polynomials and skew Schur $P$-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians. We then obtain that \[ \mathcal N(s_{\lambda/\mu}),\qquad \mathcal N(P_{\lambda/\mu}),\qquad \mathcal N(Q_{\lambda/\mu}) \] are realizable volume polynomials. This settles the skew-Schur and Schur-$P$ Lorentzian conjectures of Huh–Matherne–Mészáros–St.~Dizier and strengthens the latter to arbitrary skew $P/Q$-functions. The constructions extend to cycle transforms attached to arbitrary irreducible subvarieties of ordinary and Lagrangian Grassmannians. Their realizable-volume interpretation yields reverse Khovanskii–Teissier and Lorentzian Hodge–Riemann inequalities for ordinary and shifted tableau multiplicities; exact ordinary skew-Schur support permutahedra and extremal coefficients; the known straight shifted support polytopes with their vertex coefficients; implicit exact permutahedra for arbitrary skew $P/Q$-functions; and weighted-aggregation, covariance, and two-row Littlewood–Richardson consequences. An appendix by Zhenpeng Wang constructs the dual type~$D$ spinor cycle transform and a direct Richardson realization of $Q_{\lambda/\mu}$. In characteristic two, compatible very special isogenies induce finite flat radicial morphisms between the ambient Lagrangian and spinor models. Their restrictions to the corresponding Richardson varieties have degrees $2^{\ell(\lambda)-\ell(\mu)}$ and $2^{|\lambda|-|\mu|-\ell(\lambda)+\ell(\mu)}$, and a regular complete-intersection bridge explains the difference between the two projective-bundle shifts.

Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d^*_G(x) \ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

A set of intervals $I = \{ I_1, I_2, \dots, I_n \}$ forms a simple chain if, for every $2\leq i \leq n-1$, interval $I_i$ overlaps only with $I_{i-1}$ and $I_{i+1}$. We show that a deterministic memoryless one-directional revoking algorithm achieves a competitive ratio of $2(1 - 1/\sqrt{e}) \approx 0.786$ on the simple chain in the random order model, hence performs worse than the basic greedy algorithm without revoking that has a competitive ratio of $(1 - 1/e^2) \approx 0.864$, but better than any deterministic revoking algorithm in the adversarial model that has a competitive ratio of at most $0.75$. The proof of the latter also leads to a lower bound of $n/4$ for the advice complexity.

Classical Ehrhart theory measures the discrete capacity of a convex rational (or integral) polytope $P$ by counting the number of lattice points in the $t$-th dilate $tP$ of $P$. In this paper, we extend this paradigm by replacing a lattice point with a geometric object $Q$ of dimension at most $\dim P$. We show that the counting function $\mathrm{ehr}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\mathrm{vol}(P)t^d$, where $d = \dim P$. This result is naturally derived by induction on the dimension, based on the classical quasi-polynomiality (or polynomiality) of Ehrhart functions. Similarly, replacing $P$ with its relative interior $P^\circ$ defines $\mathrm{ehr}^\circ(P,Q;t)$, which is also shown to be an eventual quasi-polynomial (or eventual polynomial). Furthermore, we prove several formulas for relative Ehrhart functions under the condition that $P = kP_0$ and $Q = lQ_0$ for some integers $k$, $l > 0$, and polytopes $P_0$ and $Q_0$ with $\dim Q_0 > 0$ such that $Q_0$ is inscribed in $P_0$. In particular, we prove that under this condition, $\mathrm{ehr}(P,Q;-t) = (-1)^{d}\mathrm{ehr}^\circ (P,Q;t+\rho )$ $(t \gg 0)$ holds for some integer $\rho > 0$ if and only if $k \mid 2l$ via the classical Ehrhart–Macdonald reciprocity law. In addition, we also prove that under the same condition, $\mathrm{ehr}(P,Q;t) = \mathrm{ehr}^\circ (P,Q;t+\sigma )$ $(t \gg 0)$ holds for some integer $\sigma > 0$ if and only if $\mathrm{ehr}^\circ (P_0;\mathrm{codeg}\,P_0) = 1$ and $k \mid \mathrm{codeg}\,P_0$.

Let $p$ be an odd prime, let $\varnothing\neq A\subseteq\mathbb F_p$ have cardinality $N$, and let $f\in\mathbb F_p[x,y]$ be a non-degenerate quadratic polynomial. Writing $S=|A+A|$ and $M=|f(A,A)|$, we prove the full-range trade-off $S^8M^6\gtrsim N^{17}(1+N^3/p^2)^{-3}$. Consequently, $\max\{|A+A|,|f(A,A)|\}\gtrsim \min\{N^{17/14},p^{3/7}N^{4/7}\}$, and in particular the exponent $17/14$ holds throughout $N\le p^{2/3}$. The proof combines a centered collision estimate for $F(u,v,w)=f(u+v,w)$, a mixed fourth-energy bound, and a popular-sum amplification. Two complementary incidence estimates enter the argument: a centered spectral bound in the dense collision regime and a point–plane bound in the sparse regime.