Related papers: Non-Hamiltonian 2-regular Digraphs

A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence…

We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a…

A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge…

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…

A graph \textit{G} is a tuple (\textit{V}, \textit{E}), where \textit{V} is the vertex set, \textit{E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an…

Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that $1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $G^{\{k\}}$…

We prove that there exists an infinite family of 4-regular 4-connected Hamiltonian graphs with a bounded number of Hamiltonian cycles. We do not know if there exists such a family of 5-regular 5-connected Hamiltonian graphs.

A new family of strongly regular graphs, called the general symplectic graphs $Sp(2\nu, q)$, associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their…

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a…

A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every…

In contrast with Kotzig's result that the line graph of a $3$-regular graph $X$ is Hamilton decomposable if and only if $X$ is Hamiltonian, we show that for each integer $k\geq 4$ there exists a simple non-Hamiltonian $k$-regular graph…

Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ for every pair…

Trotter and Erd\"os found conditions for when a directed $m \times n$ grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an $\mathcal{O}(n^4)$ algorithm to…

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent…

We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We…

In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the…

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show…

In this note, we prove that every 4-connected optimal 2-planar graph is Hamiltonian-connected. Furthermore, we show that the 4-connectedness condition is sharp by constructing infinitely many 3-connected optimal 2-planar graphs that are…

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…

We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost…