Related papers: Improved Bounds for Codes over Trees
We introduce graph width parameters, called $\alpha$-edge-crossing width and edge-crossing width. These are defined in terms of the number of edges crossing a bag of a tree-cut decomposition. They are motivated by edge-cut width, recently…
Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…
An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors so that every color class incudes a forest and the sizes of any two color classes differ by at most one.This kind of coloring was first introduced in 2013 and can…
Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs…
We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \ge 0$ and $d>0$. We also prove necessary…
Let $C$ be a binary code of length $n$ with distances $0<d_1<\cdots<d_s\le n$. In this note we prove a general upper bound on the size of $C$ without any restriction on the distances $d_i$. The bound is asymptotically optimal.
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…
We present a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$, by asking at most $O(dn\log^2 n)$ path queries. Each path query takes as input an origin node and a target node, and…
An {\em ancestry labeling scheme} labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of…
The network coding problem asks whether data throughput in a network can be increased using coding (compared to treating bits as commodities in a flow). While it is well-known that a network coding advantage exists in directed graphs, the…
We show optimal lower bounds for spanning forest computation in two different models: * One wants a data structure for fully dynamic spanning forest in which updates can insert or delete edges amongst a base set of $n$ vertices. The sole…
We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the…
This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We…
The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…
Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent…
In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$…
We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by…
A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that $|f(u)-f(v)|\geq d + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$, where $d$ is the diameter of $G$ and $d(u,v)$ the distance…
We consider the problem of finding the smallest graph that contains two input trees each with at most $n$ vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for…