GATE CS 2008 Q42 — Graphs: Connectivity, Matching, Colouring

Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime}$ or $v \in V^{\prime}$. Let the size of the smallest vertex cover in $G$ be $k$. Let $S$ be any vertex cover of size $k$. For a vertex $v \in V$, which of the following constraints …

An undirected, unweighted, simple graph $G(V, E)$ is said to be $2$-colorable if there exists a function $c: V \rightarrow\{0,1\}$ such that for every $(u, v) \in E, c(u) \neq c(v)$. Which of the following statements about $2$-colorable graphs is/are true?

Let $G$ be an undirected graph, which is a path on $8$ vertices. The number of matchings in $G$ is $\_\_\_\_$. (answer in integer)

Consider a complete graph $K_{n}$ with $n$ vertices ( $n>4$ ). Note that multiple spanning trees can be constructed over $K_{n}$. Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two …

The number of spanning trees in a complete graph of $4$ vertices labelled $\text{A, B, C,}$ and $\text{D}$ is _________.

​​​The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$. Which of the following statements is/are always TRUE?