相关论文: A generalization of the Chv\'{a}tal-Erd\H{o}s theo…

Broersma and Veldman proved that every 2-connected claw-free and $P_6$-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and $P_6$-free graph is hamiltonian. On the other hand, Li et al.…

Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…

Let $G$ be a $k$ - connected ($k \geq 2$) graph of order $n$. If $\chi(G) \geq n - k$, then $G$ is Hamiltonian or $K_k \vee (K_k^c \cup K_{n - 2k})$ with $n \geq 2 k + 1$, where $\chi(G)$ is the chromatic number of the graph $G$.

The square of a graph G, denoted G^2, is the graph obtained from G by joining by an edge any two nonadjacent vertices which have a common neighbor. A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2,…

Let $k$ be a positive integer. A graph is said to be $(P_2 \cup kP_1)$-free if it does not contain $P_2 \cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \cup kP_1)$-free…

One of the more recent generalizations of the Erd\"os-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erd\"os-Ko-Rado property for graphs in the following manner: for a graph G and a positive integer r, G is said to…

A graph $\Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$, and connected-homogeneous (CH) if this…

Let $G$ be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is…

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest…

Dirac (1952) proved that every connected graph of order $n>2k+1$ with minimum degree more than $k$ contains a path of length at least $2k+1$. Erd\H{o}s and Gallai (1959) showed that every $n$-vertex graph $G$ with average degree more than…

A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the…

The topic is the average order $A(G)$ of a connected induced subgraph of a graph $G$. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1984, Jamison proved that the average order, over all trees of order…

Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their…

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized…

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…

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…

A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at…

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree…

A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…

Motivated by work of Erd\H{o}s, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example,…