Related papers: Large connected strongly regular graphs are Hamilt…
We provide a complete characterization of those graphons $W$ for which the inhomogeneous random graph $G(n,W)$ is asymptotically almost surely Hamiltonian. The characterization involves three conditions. Two of them constitute the…
Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph…
In this exposition, we show that a Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are smooth enough. We also provide some counterexamples…
Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected,…
In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under…
In this paper, first, we establish a sufficient condition for a bipartite graph to be Hamilton-connected. Furthermore, we also give two sufficient conditions on distance signless Laplacian spectral radius for a graph to be…
We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are…
The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper,…
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…
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph on two vertices. A graph $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. We prove that every polyhedral graph (i.e. 3-connected planar graph)…
We say that a graph $G$ on $n$ vertices is $\{H,F\}$-$o$-heavy if every induced subgraph of $G$ isomorphic to $H$ or $F$ contains two nonadjacent vertices with degree sum at least $n$. Generalizing earlier sufficient forbidden subgraph…
Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that…
The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent…
A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph…
We show that for any fixed $\alpha>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $\alpha \binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and…
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…
This paper shows NP-completeness for finding Hamiltonian cycles in induced subgraphs of the dual graphs of semi-regular tessilations. It also shows NP-hardness for a new, wide class of graphs called augmented square grids. This work follows…
A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an…
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…
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters yet the existence of many of them is still under the question. Due to this uncertainty, it is of immense interest to study their structure,…