I have seen some relation about the simple graph $G$ which is not directed.
Suppose $G$ has $n=|V|\ge1$ vertices, $m$ edges, $k$ connected components, $p$ odd cycles, $q$ even cycles. Do the following hold?
$p+q\ge 1$ then $m\ge n$
$p+q=1, k=1,$ then $ m=n$
If $p = 0$, then $G$ is bipartite.
If $q = 0$, then there are $2k$ proper colorings of $G$ using $2$ colors.
Attempts: For question i, I believe it is False as we may have many isolated vertices. for question 2 i think it is true. $p+q=1$ mean there exist cycle and k=1 mean that it is a connected graph. I believe it is true but then i have no idea how to show m=n.
for Q4, i think that it is true as it may a graph with no cycle is bipartile and also a graph with even with even cycle only seems to be a bipartitle too.
For question 5 i think it is false as for the case that the graph has no cycle, q=0 but we cna actually colour with 2 colours.
Is my interpretation or guess true? and is there any hints about the proof of some details?