Papers

Mader conjectured that every $k$-strong digraph $D$ with minimum semidegree $\delta^0(D)\ge 2k+m-1$ contains a dipath $P$ of order $m$ such that $D-V(P)$ remains $k$-strong. For $k=1$, he obtained the weaker bound $\delta^0(D)\ge 2m$. We confirm the conjecture for $k=1$ by showing that the sharp bound $\delta^0(D)\ge m+1$ suffices. As a consequence, we show that for every integer $m\ge2$, every strongly connected digraph $D$ with $\delta^0(D)\ge\max\{2,m-1\}$ contains a dipath $P$ of order $m$ such that $D-A(P)$ is strongly connected.

In 1966, Erdős, Goodman, and Pósa showed that every $n$-vertex graph $G$ admits a cover of its edges using at most $\lfloor \frac{n^2}{4}\rfloor$ cliques, with tightness witnessed by the balanced complete bipartite graph. Erdős suggested the following strengthening: every $n$-vertex graph $G$ admits an edge decomposition into cliques with total cost at most $\lfloor \frac{n^2}{4}\rfloor$, where each $i$-clique has cost $i-1$. There is another generalization conjectured by Dau, Milenkovic and Puleo: for every $t\ge4$, every $n$-vertex graph $G$ admits a cover of its $t$-cliques using at most $\prod_{j=0}^{t-1}\left\lfloor\frac{n+j}{t}\right\rfloor$ cliques. Balogh, He, Krueger, Nguyen and Wigal proved asymptotic and fractional versions of these conjectures. We solve both conjectures for large $n$ and identify the equality cases. Our proof of Erdős's conjecture combines linear programming duality and the spanning-forest polytope with a weighted stability form of Mantel's theorem, followed by explicit decomposition algorithms around an almost balanced bipartition. For the $t$-clique-cover conjecture, we combine graph removal lemma and generalized Turán stability with an exact covering construction for graphs close to $T_{n,t}$.

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ are subvarieties of the flag variety parameterized by a Hessenberg function $\mathbf{h}: [n] \to [n]$ and a matrix $\mathsf{X} \in \mathfrak{gl}_n(\mathbb{C})$. In recent work, Goldin and the second author showed the existence of flat degenerations of Hessenberg varieties to nilpotent Hessenberg varieties over the minimal sheet. This implies that all Hessenberg varieties over the minimal sheet have the same dimension. Our main result generalizes this dimension result to arbitrary sheets. Specifically, we prove that for a fixed Hessenberg function $\mathbf{h}:[n] \to [n]$, all Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ defined in the type $A$ flag variety by linear operators $\mathsf{X}$ from the same sheet of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$ have the same dimension.

For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(\sigma)=|E(G)\cap E(\sigma(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In 1977 Erdős asked whether $f(n,4)=2n-4$, observing that $K_{2,n-2}$ gives the upper bound. We prove that, for all sufficiently large $n$, \[ f(n,4)=2n-4. \] Equivalently, every sufficiently large $n$-vertex graph with at most $2n-5$ edges has a relabelling with at most three common edges. Our proof is inspired by the recent work of Fang and Hou on the Erdős–Mullin five-edge intersection problem and builds on their core–buffer and absorption framework. The main additional ingredients are a growing high-degree core $C$ satisfying \[ |C|\Delta(G-C)=o(n), \] and a rigidity analysis of the equality case in the relevant first-moment estimate. This analysis shows that the only core–buffer configuration forcing four local common edges is of $K_{2,|C|}$ type; the strict bound $e(G)\leq2n-5$ then supplies a defect which breaks this configuration.

We study Frankl's union-closed sets conjecture through the height of the inclusion poset. Working in the equivalent empty-set-free formulation, where one seeks an element contained in strictly more than half of the members, we prove the conjecture for every finite union-closed family of height at most four. Equivalently, the usual at-least-half formulation holds for every union-closed family containing the empty set and having height at most five. We also develop a structural theory for the next unresolved case. Assuming a smallest empty-set-free counterexample of height at most five, we show that it has even cardinality $2t$, at least three critical elements of frequency $t$, and satisfies the minimal-counterexample bound $t \geq 2n-1$. Every critical element determines a coatom of the form $U \setminus \{x\}$, while every critical pair satisfies a dichotomy between a full double-avoidance top and a large avoidance fiber admitting a three-layer trace normal form. Coordinate deletion further yields an exact matching-defect obstruction. Finally, introducing the minimum number of join-irreducible members required to cover all critical elements, we exclude the five-cover case and show that this critical join-cover number is either three or four. These results substantially constrain any possible height-five counterexample while leaving the remaining transfer problem explicit.

In this article, we contribute to the spectral analysis of graph associahedra by providing a lower bound for the second largest eigenvalue of $\mathcal{A}(G)$. Furthermore, using equitable partitions, we analyze the spectrum of the stellohedron $\mathcal{A}(K_{1,n})$. Specifically, we prove the existence of an eigenvalue in each interval $(n-i, n-i+1]$ for $1 \leq i \leq 5$, establish the presence of an eigenvalue with high multiplicity in $(n - \frac{3}{n} + \frac{2}{n^2-n}, n)$, and identify two additional small eigenvalues.

Related to a bounded-diameter bipartite analogue of the Henderson–Ryser conjecture, DeBiasio, Kamel, McCourt, and Sheats proved that the vertices of every $2$-colored complete bipartite graph can be covered by two monochromatic subgraphs, each of diameter at most four. We improve this bound on the diameter to the best possible value of three.

Given a set $V$ of $n$ vertices and an integer $k\ge1$, our goal is to maximize the number of edges in graphs $G_1, G_2, \ldots, G_k$, defined on $V$, under the constraint that the union of all graphs, thought of as a multi-graph, does not contain a rainbow copy of the path $P_5$ on $5$ vertices, that is, a copy of $P_5$ with each of its four edges belonging to a different $G_i$. We consider two versions of the problem, in which, respectively, $\sum_i e(G_i)$ and $\min_i e(G_i)$ is maximized. In the former case, we determine the maximum precisely for all $k\le n-1$ (and also for $P_4$). In the latter, we obtain an asymptotic value for $k\in\{5,6,9\}$ and formulate a very plausible conjecture for all other values of $k$. We also solve the problem for $k=4$, but under an additional assumption of completeness.

Standard neural architectures often fail to generalize to longer inputs for dynamic programming (DP) targets. We investigate what makes this hard geometrically. Every finite min-plus DP is a shortest path on a DAG, which is equivalently a tropical polynomial whose extended Newton polyhedron encodes the decision boundary of which path wins. We prove these three descriptions (graph, polynomial, polyhedron) form isomorphic semirings at two levels — formal polynomials and their computed functions — connected by operations that characterize all structural redundancies. We then address the length-generalization question geometrically: does the decision boundary at length $T$ decide the boundary at $T+1$? We present two structural negatives. The semiring's two native ways to reduce dimension (setting a variable to each identity) are neither injective nor always closed within the DP. Series and parallel composition fail to construct all DAG topologies from smaller sub-DAGs, and even all terminal-only operations do not capture all DP compositions.

For $N$ lattices in $\R^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $\Omega(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff \cite{kolwolff-1999Mathematika}. We close this gap by constructing, for every $d\geq 2$ and all sufficiently large $N$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering Question 1 of Kolountzakis and Papageorgiou \cite{kolPapageorgiou-functions-2022jfaa}. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.

Let $\mathcal{A}$ be a finite central real hyperplane arrangement and let $\mathcal{G}(\mathcal{A})$ be its tope graph. Koizumi recently proved that the crossing-graded magnitude homology of $\mathcal{G}(\mathcal{A})$ is torsion-free and is freely indexed by face flags. We refine his result by proving that a fixed crossing vector and terminal chamber determine a summand of rank at most one. Over $\mathbb{F}_2$, we use the canonical cohomology basis to determine the crossing-graded magnitude cohomology ring of $\mathcal{G}(\mathcal{A})$.

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.

Two competitors choose starting vertices for independent, constant-speed random walks, and each site is acquired by its first visitor. We study the spatial geometry of the resulting first-passage location game. On every finite path the optimal strategies are exactly the distributions supported on the central vertices. The proof combines reflecting-boundary harmonic barriers with a parameter-uniform aggregate estimate for product-chain exit probabilities. After diffusive rescaling, the complete two-start payoff landscape converges uniformly to the game between independent reflected Brownian motions. Its unique equilibrium concentrates at the midpoint, with explicit cubic stability. Beyond paths, attaching two leaves to every vertex of a clique of order k produces a 3k-vertex graph on which every exact optimal strategy randomizes over all k clique vertices; for k=2, this is a six-vertex tree with no pure equilibrium. The continuum best response to an endpoint is uniquely determined. A first-passage random-ranking representation relates the finite game to maximal lotteries without identifying it with nonstrategic painting, deterministic Voronoi allocation, or absorbing-trap placement. Parameter-uniform statements follow from analytic arguments or symbolic polynomial identities; identified finite exceptions and numerical enclosures have reproducible certificates.

We show that the operadic structure on the cohomology of the poset of noncrossing 2-partitions is isomorphic to the Brace operad.

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry–Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We prove that, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.

We extend the hypergraph regularity method to sparse $3$-uniform hypergraphs whose vertex links are $C_4$-free. In other words, we consider hypergraphs $H=(V,E)$ that are $K_{1,2,2}$-free, which implies that $|E|=O(|V|^{5/2})$. For such hypergraphs we establish a sparse analogue of the removal lemma for the tight cycle on seven vertices minus an edge.

We show that there exists a polynomial-time algorithm to find a stable matching in network hypergraphic preference systems. The key connection that drives the algorithm was discovered by chatting with ChatGPT-5.6 Sol Max. We verified it independently and present the details in our own words.

The crossing graph $\mathrm{Cross}(M)$ of a median graph $M$ is defined as the graph whose vertices are the $\Theta$-classes of $M$ and whose edges connect two $\Theta$-classes whenever they cross. It is known that every graph $X$ can be realised as the crossing graph of some median graph. In this article, we initiate the study of the space $\mathrm{Cross}^{-1}(X)$ of all the median graphs with crossing graph $X$. First, we prove that two finite median graphs have isomorphic crossing graphs if and only if one can be obtained from the other by a sequence of elementary transformations we call slidings. Then, motivated by the fact that $\mathrm{Cross}^{-1}(X)$ always contains a single median graph of maximal degree, namely the simplex-graph of $X$, we introduce the median degree of $X$ as the smallest possible degree of a median graph in $\mathrm{Cross}^{-1}(X)$. We compute the median degree for some families of graphs and characterise the graphs with maximal median degree.

Let $S=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with ${\rm deg} x_1=\cdots ={\rm deg} x_n = 1$ and ${\bf a} = (a_1,\ldots,a_n) \in {\mathbb Z}_{>0}^n$. Given a squarefree monomial $u=x_{i_1} \cdots x_{i_d}$ of $S$ with $1 \leq i_1 < \cdots < i_d \leq n$, we set $u^{[{\bf a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[{\bf a}]}$ with $G(I^{[{\bf a}]})=\{u^{[{\bf a}]} : u \in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[{\bf a}]}$ is studied.

Let $\mathrm{ex}(n,H)$ denote the Turán number of $H$. A graph is color-critical if there exists an edge $e\in E(H)$ such that $\chi(H-e)<\chi(H)$. For a color-critical graph $H$ with $\chi(H)=r+1$, Simonovits' chromatic critical edge theorem implies that there exists an $n_0(H)$ such that $\mathrm{ex}(n,H)=e(T_{n,r})$ and the Turán graph $T_{n,r}$ is the only extremal graph provided $n\geq n_0(H).$ Let $W_{2k+2}$ be the even wheel obtained by joining a vertex to a cycle of length $2k+1,$ where $k\geq1$ is an integer. Since $W_{2k+2}$ is color-critical and $\chi(W_{2k+2})=4$, $T_{n,3}$ is the unique extremal graph for $W_{2k+2}$-free graphs of sufficiently large $n.$ Note that the extremal graph $T_{n,3}$ is 3-partite. In this paper, we determine the exact Turán number of $W_{2k+2}$ in non-$3$-partite graphs and characterize all extremal graphs provided $n$ is sufficiently large.

Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $\delta_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $\delta_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $\delta_\chi,\delta_{\mathrm{hom}},\delta_{\mathrm{VC}}$, and $\delta_{\mathrm B}$ are pairwise distinct: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.

This paper studies the longest increasing subsequence (LIS) problem for sequences generated by dyadic-type chaotic interval maps. Starting from a single point $x\in[0,1)$ chosen uniformly at random, we form the order pattern of the first $N$ points of its orbit, with the doubling map as the basic model. Let $\lambda_1^{(N)}$ be the LIS length, equivalently the length of the first row of the Young diagram obtained by Schensted's insertion. We show that $\mathbb E[\lambda_1^{(N)}]/\sqrt N\to 2$, matching the leading asymptotics in the classical Ulam–Hammersley problem for uniform random permutations.

Identities asserting that an antisymmetrizer admits a determinantal expression are central to several proofs of counting formulas for alternating sign matrices. Antisymmetrizers appear also frequently in symmetric function theory as, for instance, the Hall-Littlewood polynomials can be realized via antisymmetrizers. In this paper, we establish a new identity of this type, thereby proving a conjecture of Lukas Riegler and one of the authors. We also elaborate on a related antisymmetrizer that may be pivotal in resolving a beautiful conjecture of Colomo and Pronko, and we formulate a version of their conjecture tailored to the suggested approach.

It is shown in this manuscript that a random graph $G$ drawn from the Erdős–Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

The spectral Turán type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recently, Xu and Chang proved that the graphs among all $n$-vertex $K_5$-free $1$-planar graphs having maximum spectral radius lie within a small family of candidates. First, this paper explicitly identifies the unique spectral extremal graph among the $n$-vertex $K_5$-free $1$-planar graphs. Second, it establishes a structural reduction theorem: For any forbidden subgraph $F$ with $\delta(F)\ge2$ that is contained in $K_2\vee P_{n-2}^{2+}$ but not in $K_2\vee I_{n-2}$, every spectral extremal $F$-free $1$-planar graph contains a spanning complete bipartite graph $K_{2,n-2}$, where $P^{2+}_{n-2}$ is obtained from a path $u_1u_2\dots u_{n-2}$ by adding edge $u_1u_{n-2}$ and all edges $u_iu_{i+2}$ for $1\le i\le n-4$, and $I_{n-2}$ denotes the empty graph on $n-2$ vertices. As applications, the graph among all $n$-vertex $C_5$-free (resp. $2C_5$-free) $1$-planar graphs having maximum spectral radius is determined. These results extend spectral Turán type problems for $1$-planar graphs from cliques to cycles and their disjoint union.

We study an operator $\Theta$ on finite integer sequences, where $\Theta(\sigma)_i$ counts the entries to the left of $\sigma_i$ that are strictly smaller than $\sigma_i$. This operator is a variant of the so-called Lehmer code. For every sequence $\sigma$, the image $\Theta(\sigma)$ is an inversion sequence, and the restriction of $\Theta$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $\Theta$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $\Theta$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.

The spectral Erdős–Ko–Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $\rho_r$ for their spectral radii. For $2\le t<k$ and $n>2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $\rho_0\ge\rho_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede–Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.

We give a complete rigorous asymptotic analysis of the generating functions of 3-noncrossing skeleton matchings and canonical 3-noncrossing skeleton diagrams. Let $F_3$ be the ordinary generating function of 3-noncrossing matchings, and let $S(y)=\sum_{n\geq 0}S(n)y^n$ be determined by $S(zF_3(z)^2)=F_3(z)$. The proof is deliberately ordered to avoid circularity. First, Lagrange inversion, a Stieltjes representation of $F_3$, exact cut-boundary estimates, and a moving horizontal Hankel contour give $S(n)\sim 24(\pi A^5)^{-1}\sigma^{-n}n^{-5}$ independently of any $\Delta$-analyticity of $S$. This estimate supplies boundary regularity of $S$ and $S'$. We then prove a global biholomorphic inversion theorem, continuation across every nonprincipal point of the convergence circle, and a logarithmically perturbed sectorial inverse theorem. A complete disk-chain and monodromy argument yields a single-valued continuation to a standard $\Delta$-domain. At the principal singularity, $S(y)=Q_4(u)-(\pi A^5)^{-1}u^4\log u+O(u^5(1+|\log u|))$, where $u=1-y/\sigma$. Finally, the canonical composition $S_3^{[4]}(z)=(1-z)(S(\vartheta(z))-1-\vartheta(z))$ is shown to be $\Delta$-analytic at its unique dominant singularity $\eta=0.49340718057613087519\ldots$, and $[z^n]S_3^{[4]}(z)\sim 7892.16205625817\ldots n^{-5}\eta^{-n}$. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.

This paper utilizes an extremely simple idea: in the multiplicative group of a finite field $F_q$, cosets of a certain subgroup are considered, and an attempt is made to combine these cosets into pairs such that the resulting induced subgraph in the corresponding Paley graph is strongly regular. It is shown that there exists an infinite sequence of Paley graphs that can be partitioned into Petersen graphs in this manner. Furthermore, a new strongly regular graph with parameters $(50, 21, 8, 9)$ is constructed using this method.