Related papers: Hamiltonicity of 3-arc graphs

We prove that every locally Hamiltonian graph with $n\ge 3$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a…

A graph $G$ is said to be $k$-$\gamma$-vertex critical if the domination numbers $\gamma(G)$ of $G$ is $k$ and $\gamma(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$\gamma_{c}$-vertex critical if the…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

A graph Gamma is said to be 2-arc-transitive if its full automorphism group Aut(\Gamma) has a single orbit on ordered paths of length 2, and for G\leq Aut(\Gamma), \Gamma is G-regular if G is regular on the vertex set of \Gamma. Let G be a…

Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that…

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in…

An oriented graph is an orientation of a simple graph. In 2009, Keevash, K\"{u}hn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, K\"{u}hn and Osthus showed that…

We study the Hamiltonian path problem in C-shaped grid graphs, and present the necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in these graphs. We also give a linear-time algorithm for…

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper,…

C. Thomassen in \cite{[11]} suggested (see also \cite{[2]}, J. C.Bermond, C. Thomassen, Cycles in Digraphs - A survey, J. Graph Theory 5 (1981) 1-43, Conjectures 1.6.7 and 1.6.8) the following conjectures : 1. Every 3-strongly connected…

Given a graph $H$, let $G^j_k(H)$ be the graph whose vertices are the proper $k$-colorings of $H$, with edges joining two colorings if $H$ contains a connected subgraph on at most $j$ vertices that includes all vertices where the colorings…

In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically…

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with…

A subgroup of the automorphism group of a graph $\G$ is said to be {\em half-arc-transitive} on $\G$ if its action on $\G$ is transitive on the vertex set of $\G$ and on the edge set of $\G$ but not on the arc set of $\G$. Tetravalent…

Yang et al. proved that every 3-connected, essentially 11-connected line graph is Hamilton-connected. This was extended by Li and Yang to 3-connected, essentially 10-connected graphs. Strengthening their result further, we prove that…

For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal…

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct…

Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of…