Related papers: On the independence number in subcubic graphs

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local…

A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a…

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

Given a finite, simple graph $G$, the independent bondage number of $G$ is the minimum size of an edge set such that its deletion results in a graph with strictly larger independent domination number than that of $G$. While the bondage…

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open…

For a given connected graph G on n vertices and m edges, we prove that its independence number is at least (2m+n+2-sqrt(sqr(2m+n+2)-16sqr(n)))/8.

A signed graph $(G, \sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, \sigma)$. Let $c(G)$, $\alpha(G)$ and $r(G, \sigma)$ be the cyclomatic number, the independence number and the rank…

A $k$-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than $k$ in the graph. The $k$-independence number of a graph, denoted $\alpha_k$, is the size of a largest…

A tree containing exactly two non-pendant vertices is called a double-star. Let $k_1$ and $k_2$ be two positive integers. The double-star with degree sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is a cubic…

The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose vertex set is $V$ and in which two distinct vertices are adjacent if and only if their distance in $G$ is at most $k$. This article proves various eigenvalue bounds…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length…

The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The {\em distinguishing…

A graph is equimatchable if each of its matchings is a subset of a maximum matching. It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint. This paper provides…

The bipartite independence number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s + t = q + 1$, such that for any two disjoint subsets…

Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…

Let v(G) and p(G) be the number of vertices and the maximum number of disjoint 3-vertex paths in G, respectively. We discuss the following old Problem: Is the following claim (P) true ? (P) if G is a 3-connected and cubic graph, then p(G) =…

The structure of all triangle free graphs G = (V,E) with |E| - 6|V| + 13\alpha(G) = 0 is determined, yielding an affirmative answer to a question of Stanis{\l}aw Radziszowski and Donald Kreher.