Related papers: Circuit diameter and Klee-Walkup constructions
A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…
A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In…
In a simple graph, a shunt is a symmetry which sends an edge to an incident edge (without fixing their shared vertex). The orbit of this edge under the shunt forms a consistent cycle. The important theorem of Biggs and Conway says that in a…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the…
In this paper, a proof of the cycle double cover conjecture is presented. The cycle double cover conjecture purports that if a graph is bridgeless, then there exists a list of cycles in the graph such that every edge in the graph appears in…
An Euler tour of a hypergraph is a closed walk that traverses every edge exactly once; if a hypergraph admits such a walk, then it is called eulerian. Although this notion is one of the progenitors of graph theory --- dating back to the…
Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random…
More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a…
In $2013$ a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil…
In 2008, Chen and Chv\'atal conjectured that in every finite metric space of $n$ points, there are at least $n$ distinct lines, or the whole set of points is a line. This is a generalization of a classical result in the Euclidean plane. The…
We study minimum degree conditions under which a graph $G$ contains $k$th powers of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of $G$ is large. This extends a result of Allen,…
Motivated by the Gilbreath conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the…
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary…
A recent generalization of the Erd\H{o}s Unit Distance Problem, proposed by Palsson, Senger and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in $3$-space. Studying a variant of…
We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high…
The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In…
The problem of whether and how one can compute the twin-width of a graph -- along with an accompanying contraction sequence -- lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at…
A strict lower bound for the diameter of a symmetric graph is proposed, which is calculable with the order $n$ and other local parameters of the graph such as the degree $k\,(\geq 3)$, even girth $g\,(\geq 4)$, and number of $g$-cycles…
Gao and Richter (1994) showed that every $3$-connected graph which embeds on the plane or the projective plane has a spanning closed walk meeting each vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995) extended…