Related papers: Uniquely K_r-Saturated Graphs

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of…

For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$ but does contain $F$ after the addition of any new edge. Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be…

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some…

A graph is K_{2,3}-saturated if it has no subgraph isomorphic to K_{2,3}, but does contain a K_{2,3} after the addition of any new edge. We prove that the minimum number of edges in a K_{2,3}-saturated graph on n >= 5 vertices is sat(n,…

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of…

Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in…

A connected r-regular graph, where $r \geq 3$, is an r-graph if each odd cut has at least r edges. Every r-graph is matching covered - a connected graph whose each edge participates in some perfect matching. We set out to: (i) characterize…

Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every…

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We…

A drawing of a graph is $k$-plane if no edge is crossed more than $k$ times. In this paper we study saturated $k$-plane drawings with few edges. This are $k$-plane drawings in which no edge can be added without violating $k$-planarity. For…

In this note, we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay. It is proved that if $G$ is an $r$-partite Cohen-Macaulay graph which is covered by some disjoint cliques of…

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number…

Graph $G$ is $F$-saturated if $G$ contains no copy of graph $F$ but any edge added to $G$ produces at least one copy of $F$. One common variant of saturation is to remove the former restriction: $G$ is $F$-semi-saturated if any edge added…

A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for…

A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\overline{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$,…

For $r \ge 2$ and a graph $G$, let $\alpha_{{r}}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $\alpha_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following…

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye…

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n]:=\{1,2,...,n\}$ and whose edge set is $\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently…

We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n,…

The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A…