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Let $G$ have a vertex cover of size at most $m$ and let the degree of $G$ be bounded by $k$. Then $G$ has at most $m(k+1)$ vertices.

Note: Remove all vertices of degree $0$.


Answer:

The idea is to have $m$ graphs like the below one.

enter image description here

But how can I prove this proposition formally?

Thank you in advance for your answers.

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    $\begingroup$ You cannot prove a property of all graphs by exhibiting a single graph. All your example shows is that $m(k+1)$ is tight, that is, there is a graph satisfying the given properties and having exactly $m(k+1)$ vertices. $\endgroup$ – Yuval Filmus Dec 18 '20 at 8:25
  • $\begingroup$ I said the main idea is to show the maximum state is like the above graph. Please read my answer again. @YuvalFilmus $\endgroup$ – Sepehr Omidvar Dec 18 '20 at 9:02
  • $\begingroup$ You need to show "if $G$ satisfies property $P$, then $G$ has at most $x$ vertices". What you're doing is "here is a graph $G_0$ satisfying property $P$ and having $x$ vertices". Sometimes you can deduce the former from the latter by showing how to transform every graph $G$ satisfying property $P$ to the graph $G_0$ without reducing the number of vertices. In this particular case, I'm not sure how to do that. In contrast, there is a very easy direct proof of your proposition. $\endgroup$ – Yuval Filmus Dec 18 '20 at 9:07
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Consider the graph $G=(V, E)$, where each vertex in $G$ has at most $k$ neighbours. Let $C\subseteq V$ be a vertex cover of size at most $m$. For a vertex $v\in C$, let $c(v) = \{v\} \cup \{ u\in V: \{u, v \}\in E\}$, that is, $c(v)$ is the set of vertices that are covered by the vertex $v$ (including $v$ itself).

A side note: the subgraph induced by the vertices in $c(v)$, for any $v$, is a "star" graph (similar to the one you attached, yet the number of vertices that are neighbours of $v$ in this subgraph can be strictly less than $k$). The idea is to consider star graphs corresponding to the vertices in $C$, and then use the union bound for sets.

Now $C$ being a vertex cover implies that $\bigcup\limits_{v\in C} c(v) = V$. This is easy, yet we prove it for completentss: the $\subseteq$-containment is trivial. The other direction follows as for every vertex $v\in V$, one of the following holds:

  • $v\in C$: in this case, as $v\in c(v)$, we have that $v\in \bigcup\limits_{v\in C} c(v)$.
  • there is $u\in C$ such that $\{u, v\}\in E$: in this case, as $v\in c(u)$, we have that $v\in \bigcup\limits_{v\in C} c(v)$.

Finally, the following holds $$ |V| = |\bigcup\limits_{v\in C} c(v)| \leq \sum\limits_{v\in C} |c(v)| \leq |C|\cdot max_{v\in C} (|c(v)|) \\\leq |C|\cdot max_{v\in C}(degree(v) + 1) \leq |C|\cdot (k + 1) \leq m\cdot(k+1)$$

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