Related papers: Strongly Perfect Claw-free Graphs -- A Short Proof
A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A {\em bull} is a graph consisting of a triangle with…
We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph…
We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $\alpha(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph.…
Let $G$ be a connected simple graph on $n$ vertices and $m$ edges. Denote $N_{i}^{(j)}(G)$ the number of spanning subgraphs of $G$ having precisely $i$ edges and not more than $j$ connected components. The graph $G$ is \emph{strong} if…
Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest…
In this paper, we study graphs whose matching polynomial have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs.. We show that…
A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and…
Let $G$ be a bipartite graph, and let $H$ be a bipartite graph with a fixed bipartition $(B_H,W_H)$. We consider three different, natural ways of forbidding $H$ as an induced subgraph in $G$. First, $G$ is $H$-free if it does not contain…
For a positive integer $k$, we say that a graph is $k$-existentially complete if for every $0 \leq a \leq k$, and every tuple of distinct vertices $x_1,\ldots,x_a$, $y_1,\ldots,y_{k-a}$, there exists a vertex $z$ that is joined to all of…
Let $G$ be a graph on $n$ vertices. For $i\in \{0,1\}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even…
This paper proves a conjecture generated by the artificial intelligence conjecturing program called \emph{TxGraffiti}. More specifically, we show that if $G$ is a connected, cubic, and claw-free graph, then $Z(G) \le \gamma(G) + 2$, where…
A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…
Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative…
A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$,…
In 1962, Erd\H{o}s proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $\delta(G)\geq k$…
A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural…
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove…
A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induced odd cycle on at least 5 vertices. A graph is perfect if the chromatic number equals the maximum clique number for every induced subgraph.…
A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least $5$. In this paper, we extend this result by providing a…