SIAM Journal on Discrete Mathematics

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The minimum positive co-degree of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is ...

The Kimura 3-parameter model is one of the most fundamental phylogenetic models in algebraic statistics. We prove that all algebraic varieties associated to this model are projectively normal, confirming a conjecture of Michałek.

For an $r$-graph $G$, the minimum $(r-1)$-degree $\delta(G)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(G)$ is contained in at least $t$ edges of $G$. Given an $r$-graph $F$, the codegree density $\gamma(F)$ is the largest $\gamma>0$ ...

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we ...

Given an $n$-vertex graph $H$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $H \cup \mathbb{G}(n,p)$ over the supergraphs of $H$ is referred to as a (random) perturbation of $H$. We consider the distribution of edge-colored ...

In the present paper, we consider the majorization theorem (also known as Karamata's inequality) and the respective minima of the majorization (the so-called $M$-sets) for $f$-energy potentials of $m$-point configurations on the unit sphere. In particular, ...

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $...

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove ...

Lower bounds for the Rényi entropies of sums of independent random variables taking values in cyclic groups of prime order under permutations are established. The main ingredients of our approach are extended rearrangement inequalities in prime cyclic ...

In the literature there are several determinant formulas for Schur functions: the Jacobi--Trudi formula, the dual Jacobi--Trudi formula, the Giambelli formula, the Lascoux--Pragacz formula, and the Hamel--Goulden formula, where the Hamel--Goulden formula ...

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit longest path transversals of constant size. ...

In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the nonempty subsets $A\subset \{1,\dots,n\}$, is it possible to construct a body $T\subset \mathbb{R}^n$ such that $x_A=|T_A|$, where $|T_A|$ is the $|A|$-...

We show that for any regular matroid on m elements and $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid, is bounded by $m^{O(\alpha^2)}.$ ...

Motivated by a conjecture of Grünbaum and a problem of Katona, Kostochka, Pach, and Stechkin, both dealing with non-Hamiltonian $n$-vertex graphs and their $(n-2)$-cycles, we investigate $K_2$-Hamiltonian graphs, i.e., graphs in which the removal of any ...

A 1-planar graph is a graph that can be drawn in the Euclidean plane such that each edge crosses at most one edge. An independent crossing (IC)-planar graph is a 1-planar graph satisfying the condition that two pairs of crossing edges have no common end-...

Motivated by the problem of bounding the number of iterations of the simplex algorithm, we investigate the possible lengths of monotone paths followed inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the ...

For each integer $k\ge 2$, we determine a sharp bound on ${mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$, where $I$ is an independent set and $G[F_k]$ is a forest in which each component has at most $k$ vertices. For each $k$ we ...

As is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical two-dimensional Weisfeiler--Leman algorithm (2-WL). On the other hand, the prominent Cai--Fürer--Immerman construction shows ...

We study the Hamiltonicity of the following model of a random graph. Suppose that we partition $[n]$ into $V_1,V_2,\ldots,V_k$ and add edge $\{x,y\}$ to our graph with probability $p$ if there exists $i$ such that $x,y\in V_i$. Otherwise, we add the edge ...

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and ...

The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It is known that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the 3-uniform hypergraph on $...

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, and let ${\mathcal S}$ denote the set of all nonnegative integer solutions of the equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots +y_mb_m$. A solution $(x_1,\ldots,x_n,y_1,\ldots,y_m)$ in ...

A 0-1 matrix $M$ is saturating for a 0-1 matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by changing some 1 entries to 0 entries, and changing an arbitrary 0 to 1 in $M$ introduces such a submatrix in $M$. In saturation problems ...

We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ facets, ...

Given a graph $G$, a $q$-open neighborhood conflict-free coloring or $q$-ONCF-coloring is a vertex coloring $c\colon V(G) \rightarrow \{1,2,\ldots,q\}$ such that for each vertex $v \in V(G)$ there is a vertex in $N(v)$ that is uniquely colored from the ...

For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$) of circumference $\ge k$ has a cycle whose length is in ...

We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials and, more generally, the $q$-Schröder polynomials. By interpreting them as $\mathfrak{sl}_2$-characters, we show that the rational $q$-...

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial ${P}$ of degree at most $d$ that agrees with $f$ at each point with high probability. Introduced by ...

For every directed acyclic graph $G$, we characterize the faces of the root polytope $\tilde Q_G = \textup{conv}\{0, e_i - e_j : (i,j) \in E(G)\}$ combinatorially. Our results specialize to state of the art results in a straightforward way.

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their ...