Related papers: TSP Tours in Cubic Graphs: Beyond 4/3

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…

This paper gives simple distributed algorithms for the fundamental problem of computing graph distances in the Congested Clique model. One of the main components of our algorithms is fast matrix multiplication, for which we show an…

A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for $3$-edge-colourable cubic graphs, but remains widely open for…

Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a…

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if…

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t$. In 2024, Zhan conjectured that every $2$-connected $[p + 2, p]$-graph of order at least $2p + 3$ and with minimum degree at least $p$…

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph…

Given a connected undirected graph $\bar{G}$ on $n$ vertices, and non-negative edge costs $c$, the 2ECM problem is that of finding a $2$-edge~connected spanning multisubgraph of $\bar{G}$ of minimum cost. The natural linear program (LP) for…

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In Borgs…

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning trail with maximum…

Let $G$ be a graph and $T$ a certain connected subgraph of $G$. The $T$-structure connectivity $\kappa(G; T)$ (or resp., $T$-substructure connectivity $\kappa^{s}(G; T)$) of $G$ is the minimum number of a set of subgraphs…

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2…

Given a complete graph $K_{n}=(V, E)$ with non-negative edge costs $c\in {\mathbb R}^{E}$, the problem $2EC$ is that of finding a 2-edge connected spanning multi-subgraph of $K_{n}$ of minimum cost. The integrality gap $\alpha\text{2EC}$ of…

How many graphs on an $n$-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly $1/2^{n-1}$ of all graphs. Our aim in this short…

Let $\mathcal{G}(k)$ denote the set of connected $k$-regular graphs $G$, $k\geq2$, where the number of vertices at distance 2 from any vertex in $G$ does not exceed $k$. Asratian (2006) showed (using other terminology) that a graph…

We give a new, strongly polynomial-time algorithm and improved analysis for the metric $s-t$ path TSP. It finds a tour of cost less than 1.53 times the optimum of the subtour elimination LP, while known examples show that 1.5 is a lower…

The location-domination number is conjectured to be at most half of the order for twin-free graphs with no isolated vertices. We prove that this conjecture holds and is tight for subcubic graphs. We also show that the same upper bound holds…

In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with…

We prove that every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian, and show that the 6-face condition is tight. Our results push the connectivity condition of the Barnette-Goodey conjecture to the weakest…

The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements.…