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I don't understand this that my professor said.

Let $G$ be a bipartite graph with max degree $d$ and $m$ edges.

Then,

let$ X$ be a vertex cover of $G$.

As every vertex of X is incident to at most $d$ edges, and every edge of $G$ is incident to a vertex in $X$, $d|X| \ge m$

I don't get how the inequality was so quickly gained from that statement.

Can someone please help to explain this

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    Go back to the definition of "vertex cover." – 2017-02-02
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    The definition we were given was . A vertex cover is a set of vertices such that each edge is incident to atleast one vertex in the cover – 2017-02-02

1 Answers 1

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Let $T$ be the set of all pairs $(x,e)$ where $x\in X$ and $e$ is an edge containing $x$ as an endpoint.

We know by the definition of a vertex cover that for each $e$, there is an $x\in X$ so that $(x,e)\in T$. Thus $|T|\geq m$. There are at least $m$ different elements in $T$.

Bur $|T|\leq d|X|$, because each node $x\in X$ is on at most $d$ edges.

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    Now I understand – 2017-02-02