Question
Consider the statement below-:
$\text{1.In simple graph with 6 vertices,if degree of each vertex is 2 ,then graph is connected}$
$\text{2.In simple graph with 6 vertices,if degree of each vertex is 2 ,then graph is Euler}$
$\text{3.In simple graph ,if degree of each vertex is 3 ,then graph is connected}$
My Approach
$1.\text{Using Handshaking lemma,}$
$2+2+2+2+2+2=2 \times |Edges| $
$\text{number of Edges}=6$
Thus a simple graph having $6$ vertices and $6$edges and each having degree $2$ will be connected because it will also be a simple cycle.
$2.$As graph is connected and degree of each vertex is even ,thus it will be a Eulerian graph
$3.$ If degree of each vertex is $3$ , then we cannot say anything about its connectivity. It may be connected or disconnected.
Am i correct? I am not sure about $3$. Please help
Answer to all cases are "not necessrily". An example for the first two questions is a disjoint union of two triangles $K_3$, and for the third a disjoint union of two tetrahedra $K_4$.