Combinatorics (math.CO)

Abstract: Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets and let $\Delta(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The \emph{diversity} of $\mathcal{F}$ is defined as $d(\mathcal{F}) := |\mathcal{F}| - \Delta(\mathcal{F})$. Diversity can be viewed as a measure of distance from the `trivial' maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed, the diversity of this family is $0$. Moreover, the diversity of the largest non-trivial intersecting family \`a la Hilton-Milner is $1$. It is known that the maximum possible diversity of an intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is $\binom{n-3}{r-2}$ as long as $n$ is large enough. We introduce a generalization called the \emph{$C$-weighted diversity} of $\mathcal{F}$ as $d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot \Delta(\mathcal{F})$. We determine the maximum value of $d_C(\mathcal{F})$ for intersecting families $\mathcal{F} \subseteq \binom{[n]}{r}$ and characterize the maximal families for $C\in \left[0,\frac{7}{3}\right)$ as well as give general bounds for all $C$. Our results imply, for large $n$, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl's Delta-system method.

Abstract: Let $q$ be a prime power. We study extremal Peisert-type graphs of order $q^2$ without the strict-EKR property, that is, Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges. First, we determine this minimum number of edges for each value of $q$. If $q$ is a square, we show the uniqueness of extremal graph and its isomorphism with certain affine polar graph. Using the isomorphism, we conclude that there is no Hilton-Milner type result for this extremal graph. We also prove the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. If $q$ is a cube but not a square, we show the uniqueness of extremal graph and determine the number and the structure of non-canonical cliques. Finally, we show such uniqueness result does not extend to all $q$.

Abstract: The alternating sign matrices-descending plane partitions (ASM-DPP) bijection problem is one of the most intriguing open problems in bijective combinatorics, which is also relevant to integrable combinatorics. The notion of a signed set and a signed bijection is used in [Fischer, I. \& Konvalinka, M., Electron. J. Comb., 27 (2020) 3-35.] to construct a bijection between $\text{ASM}_n \times \text{DPP}_{n-1}$ and $\text{DPP}_n \times \text{ASM}_{n-1}$. Here, we shall construct a more natural alternative to a signed bijection between alternating sign matrices and shifted Gelfand-Tsetlin patterns which is presented in that paper, based on the notion of compatibility which we introduce to measure the naturalness of a signed bijection. In addition, we give a bijective proof for the refined enumeration of an extension of alternating sign matrices with $n+3$ statistics, first proved in [Fischer, I. \& Schreier-Aigner, F., Advances in Mathematics 413 (2023) 108831.].

Abstract: We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and Garc\'ia-Col\'in, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer.

Abstract: An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and related fields.

Abstract: Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and $E_1, \dots, E_k$ are subsets of $\mathbb{F}_p$, there exist linear forms $\psi_1, \dots, \psi_C: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ and subsets $F_1, \dots, F_C$ of $\mathbb{F}_p$ such that the set $U=\{x \in S^n: \psi_1(x) \in F_1, \dots, \psi_C(x) \in F_C\}$ is contained inside the set $V=\{x \in S^n: \phi_1(x) \in E_1, \dots, \phi_k(x) \in E_k\}$, and the difference $V \setminus U$ has density at most $\epsilon$ inside $S^n$. We then generalize this result to one where $\phi_1, \dots, \phi_k$ are replaced by homomorphisms $G^n \to H$ for some pair of finite Abelian groups $G$ and $H$, and to another where they are replaced by polynomial maps $\mathbb{F}_p^n \to \mathbb{F}_p$ of small degree.

Abstract: For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that contains one of $e_1$ and $e_2$ (resp. $v_1$ and $v_2$) but not the other. We determine the size of a minimum edge-separating path system of an arbitrary tree $T$ as a function of its number of leaves and degree-two vertices. We obtain bounds for the size of a minimal vertex-separating path system for trees, which we show to be tight in many cases. We obtain similar results for a variation of the definition, where we require the path system to separate edges and vertices simultaneously. Finally, we investigate the size of a minimal vertex-separating path system in Erd\H{o}s--R\'enyi random graphs.

Abstract: In this paper we determine all fusions of the association scheme $\mathcal{A} \otimes \mathcal{A}$, where $\mathcal{A}$ is the symmetric rank $3$ association scheme corresponding to a strongly regular graph. This includes both guaranteed fusions, which are fusions for all symmetric rank $3$ association schemes $\mathcal{A}$, and specific case fusions, which only exist under restrictions on the parameters of the association scheme. Along the way we will determine the fusions of wreath products of strongly regular graphs and the fusions of the tensor square of a symmetric rank $3$ table algebra. This extends recent work of the authors and Meagher, which solved the same problem for the generalized Hamming scheme $H(2,\mathcal{A})$ of the association scheme obtained from a strongly regular graph. The main results of this article show (1) the families of strongly regular graphs for which $\mathcal{A} \otimes \mathcal{A}$ has a special case fusion are the same families for which $H(2,\mathcal{A})$ has a special case fusion; and (2) the imprimitive strongly regular graphs are the only family of strongly regular graphs for which the wreath product $\mathcal{A} \wr \mathcal{A}$ has a special case fusion.