SIAM Journal on Discrete Mathematics

For a positive integer $k$, a book (with $k$ pages) is a topological space consisting of a spine, which is a line, and $k$ pages, which are half-planes with the spine as their boundary. We say that a graph $G$ admits a $k$-page book embedding or is $k$-...

Compared to the classical binomial random (hyper)graph model, the study of random regular hypergraphs is made more challenging due to correlations between the occurrence of different edges. We develop an edge-switching technique for hypergraphs which ...

The representation complexity of a bipartite graph $G=(P,Q)$ is the minimum size $\sum_{i=1}^s (|A_i|+|B_i|)$ over all possible ways to write $G$ as a (not necessarily disjoint) union of complete bipartite subgraphs $G=\cup_{i=1}^s A_i\times B_i$ where $...

The family of judicious partitioning problems, introduced by Bollobás and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance ...

Let $C^t$ denote the loose cycle on $t = 2s$ vertices, that is, the 3-uniform hypergraph obtained from a graph cycle $C$ on $s$ vertices by replacing each edge $e = \{u, v\}$ of $C$ with the edge triple $\{u, x_e, v\}$, where $x_e$ is uniquely assigned ...

In a seminal paper, Chen, Roughgarden, and Valiant [SIAM J. Comput., 39 (5) (2010), pp. 1799--1832] studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-...

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many ...

Mubayi and Pikhurko established several Turán-type results and stability results for $r$-uniform hypergraphs. In particular, they considered hypergraphs that avoid a copy of an expanded complete 2-graph and a copy of a Fan-hypergraph. Their Turán ...

An important result of Komlós [Tiling Turán theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x \in (...

In this paper we develop new techniques for covering linear programs and covering integer linear programs to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is ...

We consider the problem of finding a subcomplex $\mathcal{K}'$ of a simplicial complex $\mathcal{K}$ such that $\mathcal{K}'$ is homeomorphic to the 2-dimensional sphere, $\mathbb{S}^2$. We study two variants of this problem. The first asks if there ...

A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone if it flips between 0 and 1 at most $k$ times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in ...

The notion of digits in finite fields was introduced a few years ago as an attempt to deliver insight in related yet unresolved questions over the natural numbers. Several such intractable questions are related to the sum of digits function, which ...

We consider the enumeration of walks on the nonnegative lattice $\mathbb{N}^{d},$ with steps defined by a set $\mathcal{S}\subset \{-1, 0, 1\}^d\backslash\{{0}\}$. Previous work in this area has established asymptotics for the number of walks in certain ...

Given a dense subset $A$ of the first $n$ positive integers, we provide a short proof showing that for $p=\omega(n^{-2/3}),$ the so-called randomly perturbed set $A \cup [n]_p$ a.a.s. has the property that any 2-coloring of it has a monochromatic Schur ...

We show that in the random hyperbolic graph model as formalized by Gugelmann, Panagiotou, and Peter (2012) in the most interesting range of $\frac12 < \alpha < 1$ the size of the second largest component is $\Theta((\log n)^{1/(1-\alpha)})$. Our research ...

A graph $K$ is multiplicative if a homomorphism from any product $G \times H$ to $K$ implies a homomorphism from $G$ or from $H$. Hedetniemi's conjecture stated that all cliques are multiplicative. In an attempt to explore the boundaries of current ...

Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let $f^k_\ell(n,m)$ be the minimum value of $\Delta(H)$ over all $P^k_\ell$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In this paper we study the behavior of $f^4_2(n,m)$ and $f^3_3(...

We revisit the classic problem of estimating the moments of the degree distribution of an undirected simple graph. Consider an undirected simple graph $G=(V,E)$ with $n$ (nonisolated) vertices, and define (for $s > 0$) $M_s= \sum_{v \in V} d^s_v$. Our ...

We use $P$-tableaux to give a combinatorial proof of the $e$-positivity of chromatic quasi-symmetric functions with bounce number two and some of those with bounce number three, which enables us to give a combinatorial model of the coefficients in the $e$...

We show that the expected size of the maximum agreement subtree of two $n$-leaf trees, uniformly random among all trees with the shape, is $\Theta(\sqrt{n})$. To derive the lower bound, we prove a global structural result on a decomposition of rooted ...

We consider the following natural “above-guarantee” parameterization of the classical Longest Path problem: For given vertices $s$ and $t$ of a graph $G$ and integer $k$, the Longest Detour problem asks for an $(s,t)$-path in $G$ that is at least $k$ ...

We study the problem of determining sat$(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two nonadjacent vertices from different parts ...

For graphs $H$ and $F$, let ${ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalizes the well-studied Turán numbers of graphs, was initiated recently by Alon and ...

We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any vertex by the rotor walk is at most equal to the expected ...

Gelfand--Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand--Tsetlin polytope ${GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well ...

An edge coloring of a graph $G$ is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai $r$-colorings of $K_n$ is $(\binom{r}{2}+o(1))2^{\binom{n}{2}}$. This result indicates that almost all Gallai $r$-colorings of $K_n$...

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. This conjecture is famous as one of a hundred unsolved problems selected in [A. Bondy ...

We consider the integrality gap of the subtour linear program (LP) relaxation of the traveling salesman problem (TSP) restricted to circulant instances. De Klerk and Dobre [Discrete Appl. Math., 159 (2011), pp. 1815--1826] conjectured that the value of ...

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by Lemmens and Seidel; namely, we use linear algebra and combinatorial ...