Related papers: On a conjecture of Faudree and Schelp

Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Guti\'errez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon…

It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83--90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper, we improve this bound…

The $W_v$-Path Conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee proved that the…

Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one…

A graph $G$ is \textit{asymmetric} if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi in 1963. They showed that the probability of a graph on $n$ vertices being asymmetric tends to…

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can…

Lov\'{a}sz conjectured that every connected vertex-transitive graph contains a hamilton path in 1970. First we reveal the structure of connected vertex-transitive graphs with an odd number of vertices. Then we prove that every connected…

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ…

For any undirected and simple graph G = (V;E), where V denotes the vertex set and E the edge set of G. G is called hamiltonian if it contains a cycle that visits each vertex of G exactly once. Ore (1960) proved that G is hamiltonian if…

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and…

Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that…

Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces…

We find the Ramsey number of a cycle vs. a complete graph when the order of the cycle is at least 4 times as large as the order of the complete graph. This partially confirms a conjecture of Erd\H{o}s, Faudree, Rousseau, and Schelp made in…

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several…

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lov\'{a}sz (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we…

A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much…

A graph $G$ is called an $L_1$-graph if $d(u)+d(v)\ge|N(u)\cup N(v)\cup N(w)|-1$ for every triple of vertices $u,v,w$ where $u$ and $v$ are at distance 2 and $w\in N(u)\cap N(v)$. Asratian et al. (1996) proved that all finite connected…

The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most…

We prove that, for large $n$, every $3$-connected $D$-regular graph on $n$ vertices with $D \geq n/4$ is Hamiltonian. This is best possible and confirms a conjecture posed independently by Bollob\'as and H\"aggkvist in the 1970s. The proof…

In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s_1,t_1}, ..., {s_k,t_k} in G, not necessarily distinct, there are…