SIAM Journal on Discrete Mathematics

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumor in a graph $G$, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of $G$, one to each vertex. Initially, one vertex of ...

A total-weighting of a graph $G=(V,E)$ is a mapping $f$ which assigns to each element $y\in V\cup E$ a real number $f(y)$ as the weight of $y$. A total-weighting $f$ of $G$ is proper if the coloring $\phi_{f}$ of the vertices of $G$ defined as $\phi_{f}(v)=f(...

We show two results related to finding trees and paths in graphs. First, we show that in $O^*(1.657^k2^{l/2})$ time one can either find a $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph or conclude that such a tree does not exist. Our ...

A loopless digraph is a tournament if for every pair of distinct vertices $u$ and $v$, exactly one of $uv$ and $vu$ is an edge. For a tournament $T$, a set $S\subseteq V(T)$ is homogeneous if every vertex $v$ outside of $S$ is either complete to $S$ or ...

Let $\chi_{\ell}$ and $\chi_{OL}$ denote the list-chromatic number and online list-chromatic number. We prove that if $G$ is a quasi-line graph with maximum degree greater than clique number, i.e., $\Delta(G)>\omega(G)$, and $\Delta(G)\ge 69$, then its ...

We show that for any lattice $\mathcal{L} \subseteq \mathbb{R}^n$ and vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, $\rho(\mathcal{L} + \mathbf{x})^2 \rho(\mathcal{L} + \mathbf{y})^2 \leq \rho(\mathcal{L})^2 \rho(\mathcal{L} + \mathbf{x} + \mathbf{y}) \...

In this paper, using the technique of the divided difference operators, we introduce a $4n$-point elliptic interpolation formula. Some applications of our interpolation formula to elliptic and basic hypergeometric series are given.

In this paper, we show the Erdös--Pósa property for edge-disjoint packing of $S$-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph $G$ and any vertex subset $S$, either $G$ has $k$ ...

Given a group-based Markov model on a tree, one can compute the vertex representation of a polytope describing a toric variety associated with the algebraic statistical model. In the cases of ${\mathbb Z}_2$ and ${\mathbb Z}_2\times{\mathbb Z}_2$, these ...

For any positive integer $d$, a subset $A$ of ${\mathbb Z}_{m}$ is called a basis of order $d$ for ${\mathbb Z}_{m}$ if every element of ${\mathbb Z}_{m}$ can be written as a sum of at most $d$ not necessarily distinct elements of $A$, where ${\mathbb Z}_{...

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings ...

Split networks are a popular tool for the analysis and visualization of complex evolutionary histories. Every collection of splits (bipartitions) of a finite set can be represented by a split network. Here we characterize which collection of splits can ...

Given a set of red and blue points in the plane, a bichromatic line is a line containing at least one red and one blue point. We prove the following conjecture of Kleitman and Pinchasi. Let $P$ be a set of $n$ red, and $n$ or $n-1$ blue points in the plane. ...

We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special ...

Given a distance matrix consisting of pairwise distances between species, a distance-based phylogenetic reconstruction method returns a tree metric or equidistant tree metric (ultrametric) that best fits the data. We investigate distance-based phylogenetic ...

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at least five vertices, and the lower bound is best possible. A natural question arises of whether we can improve the lower bound by changing the ...

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We derive new upper bounds on the cardinality of equiangular lines. Let us denote the maximum cardinality of equiangular lines in $\mathbb{R}^n$ with ...

We prove that, for any positive $q$, the $q$-Eulerian polynomial of type $D$ has only real zeros. This settles an open problem of Brenti in 1994. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which was ...

In this paper we study properties and invariants of matrix codes endowed with the rank metric and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper ...

In a series of four papers we prove the following relaxation of the Loebl--Komlós--Sós conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac{1}{2}+\alpha)n$ vertices of degree ...

This is the second of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac{1}{2}...

This is the third of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac12+\...

This is the last of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac12+\...

Let $G$ be a connected general graph. For any vertex $v\in V(G)$ and any function $f: V(G)\to\mathbb{Z}^+$, we introduce a set $J_f^*(v)$ consisting of the integer $f(v)$ and all odd integers less than $f(v)$, including all negative odd integers. In this ...

Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random $k$-SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, ...

We study vertex-labeled graphs that can be embedded on a given point set such that every edge is a polyline with $k$ bends per edge, where $k\in \mathbb{N}$. It is shown that on every $n$-element point set in the plane, at most $\exp(O(n\log(2+k)))$ ...

We provide strongly polynomial time combinatorial algorithms to minimize the largest eigenvalue of the weighted Laplacian of a bipartite graph and the weighted signless Laplacian of an arbitrary graph by redistributing weights among the edges. This is ...

The notion of resolving sets in a graph was introduced by Slater [Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Util. Math., Winnipeg, 1975, pp. 549--559] and Harary and Melter [Ars Combin., 2 (1976), pp. ...

A valued constraint satisfaction problem (VCSP) instance $(V,\Pi,w)$ is a set of variables $V$ with a set of constraints $\Pi$ weighted by $w$. Given a VCSP instance, we are interested in a reweighted subinstance $(V,\Pi'\subset \Pi,w')$ that preserves ...

Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, $\mathsf{CAN}(t,k,v)$, in a covering array for given values of the parameters ...