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Having an exercise from an old book of graph theory that does not contain the solution. My graph theory skills are at best entry-level. Need the concept of this problem to be explained(if possible) not the solution.

Having a simple non-directed graph that contains cycles of odd length but no $3$-length cycle. If $C$ a minimal cycle of odd (no 3) length of the graph then show:

1)if $x∈V(G) - V(C)$ and $u,w∈V(C)$, with $u,w$ being adjacent to $x$, the vertices $u,w$ are connected by a path length 2 that are made from edges of cycle c.

2)every vertex of $G$ which doen't belong to $V(C)$ have at most 2 neighborhoods in $V(C)$.

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Hints:

1) Clearly $u, w$ are connected in the cycle. It's a cycle, after all. Let's say they aren't connected by a length-2 path in $C$. Then the shortest path between them in $C$ is either of length $1$ or of length more than $2$. Show that either of these cases leads to some contradiction.

2) Take an $x\notin V(C)$ with $3$ distinct neighbours $u, v, w\in V(C)$. Use the result of 1) to discuss what $C$ must look like and again reach a contradiction.

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  • $\begingroup$ Your answer is very helpful at least for the 1st, and i am thinking i am close at solving it, so the maximum favorable distance between u,w can be n-1/2 (n=vertices), the connection between x,u,w creates a cycle which if odd is smaller than the C which can't be done (since C is the smallest one), but what if length is 6 and the distance of path between u,w is 3(or more)? $\endgroup$ – ChonChon Mar 25 at 21:02
  • $\begingroup$ @ChonChon If the shortest path from $u$ to $w$ in $C$ has length $1$, then what can you say about $xuwx$? If the shortest path in $C$ from $u$ to $w$ is 3 or longer, then one of the two paths from $u$ to $w$ in $C$ has even length ($C$ has odd length and the length of $C$ is the sun of the lengths of the two paths). Can you use $C$ and $uxw$ to make an odd cycle shorter than $C$? Finally, for 2), by 1), there must be a path in $C$ of length 2 from $u$ to $v$, and such a path from $v$ to $w$, and such a path from $w$ to $u$. These three paths together make a cycle of length $6$ contained in $C$ $\endgroup$ – Arthur Mar 25 at 22:44

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