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Let $G$ be an even graph. Show that $c(G-v)\leq \frac{d(v)}{2}$, $\forall v \in V(G)$.

Where $c(G)$ is the number of connected components of $G$ and $d(v)$ is the degree of $v$.


If $G$ is even then there is an Euler tour in $G$. Hence, any edge in $G$ must be contained in a cycle, and because of that cannot be cut edge.

I know that $G$ still may hold cut vertexes, but what can I say about them ? I know that since $v$ is in an Euler tour then there are $2$ edges inciding in $v$ for each cycle that contains it. So, $G-v$ has $2d(v)$ less edges than $G$, but only $d(v)$ connected components. I.e. you have to remove two edges before disconnecting a cycle. Is that a proof?

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You don't say it, but it is required that $G$ is connected.

The Euler circuit is the key to your proof. Take an example of an Euler circuit in $G$ starting at $v$, and this trail revisits $v$ exactly $d(v)/2$ times (including the terminal visit). Thus removing $v$ breaks that trail into $d(v)/2$ trails, each of which must be confined to a connected component.

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  • $\begingroup$ Is an Euler circuit the same as an Euler tour? $\endgroup$ Nov 28 '17 at 18:05
  • $\begingroup$ A circuit starts and ends at the same node, otherwise yes. A connected even graph (all vertices are of even degree) always has such a circuit. $\endgroup$
    – Joffan
    Nov 28 '17 at 20:55

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