Related papers: Star and Semi-Star Transformations in Fullerene Gr…

A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of consistent sizes such that $MP = PS$. Perfect structures comprise similar matrices, eigenvectors, perfect colorings (equitable partitions) and graph coverings. In this…

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ is denoted by $H(t)$ and it is defined by $H(t):=\exp{\left(itA\right)},\;t\in\mathbb{R}.$ The graph $G$ has perfect state transfer (PST) from a vertex $u$ to…

Ozkan et al. conjectured that any packing of $n$ spheres with generic radii will be stress-free, and hence will have at most $3n-6$ contacts. In this paper we prove that this conjecture is true for any sphere packing with contact graph of…

For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO $\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in a graph in…

Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $\gamma(G)$, $\gamma_{\rm pr}(G)$, and $\gamma_{t}(G)$, respectively. For…

In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph $G$, we say that a plane subgraph $H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge (possibly with…

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…

In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the…

A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to…

Given a graph $G$, the maximum size of an induced subgraph of $G$ each component of which is a star is called the edge open packing number, $\rho_{e}^{o}(G)$, of $G$. Similarly, the maximum size of an induced subgraph of $G$ each component…

A factor of a graph is a spanning subgraph. Spectral sufficient conditions are provided via spectral radius and signless Laplacian spectral radius for graphs with (i) a matching of given size (particularly, $1$-factor) containing any given…

Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph ${\Gamma}(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that ${\Gamma}(G)$ is strong $k$ star free if…

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is 0-super positive if…

Perfect state transfer in graphs is a concept arising from quantum physics and quantum computing. Given a graph $G$ with adjacency matrix $A_G$, the transition matrix of $G$ with respect to $A_G$ is defined as $H_{A_{G}}(t) =…

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any…

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the…

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a…

A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars…

A spectral characterization of the matching number (the size of a maximum matching) of a graph is given. More precisely, it is shown that the graphs G of order n whose matching number is k are precisely those graphs with the maximum skew…

A packing of two $k$-uniform hypergraphs $H_1$ and $H_2$ is a set $\{H_1', H_2'\}$ of edge-disjoint sub-hypergraphs of the complete $k$-uniform hypergraph $K_n^{(k)}$ such that $H_1'\cong H_1$ and $H_2'\cong H_2$. Whilst the problem of…