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I'm teaching an undergraduate combinatorics class, using Harris et al.'s book ``Combinatorics and Graph Theory''. In Section 1.6 there is an exercise asking to show that for the complement of a bipartite graph, the chromatic number equals the clique number. I assigned the problem to my students, without thinking much about the solution.

Now that I've given it some thought, I've found what seems to be a very natural proof using Hall's marriage theorem, and have found other proofs online that use the K\"onig-Egerv\'ary theorem. Unfortunately, my students don't know either of these results ... they don't appear until Section 1.7 of the book.

My question is this: is there a way of showing (directly, i.e., not using Lov\'asz's perfect graph theorem) that $\chi(\overline{G})=\omega(\overline{G})$ for bipartite $G$, that avoids Hall's theorem or the K\"onig-Egerv\'ary theorem? In particular, is there a way that might be found by a student unfamilar with these results, who has only seen the basics of coloring (definition of $\chi$, greedy algorithm, Brooks' theorem, some basic bounds), and knows nothing yet about perfect graphs and the perfect graph theorem?

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Hmm. It is easy to see that $\chi\left(\overline G\right) = n - \left(\text{number of edges in a maximum matching of }G\right)$ and $\omega\left(\overline G\right) = \left(\text{size of maximal independent subset of }G\right) = n - \left(\text{size of minimal vertex cover of }G\right)$ (because independent subsets are exactly the complements of vertex covers). So this exercise doesn't just follow from König's theorem; it is also equivalent to it... –  darij grinberg Feb 25 '12 at 3:50
    
PS: By "equivalent", I mean "equivalent by an argument substantially simpler than any proof I know for König's theorem". –  darij grinberg Feb 25 '12 at 3:51
    
Thanks, this is a very helpful response! –  David Galvin Feb 25 '12 at 3:58
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1 Answer

up vote 1 down vote accepted

According to Wikipedia
http://en.wikipedia.org/wiki/König's_theorem_(graph_theory)#Connections_with_perfect_graphs
the statement that $\chi(\overline{G}) = \omega(\overline{G})$ for all bipartite $G$ is actually equivalent to König's Theorem.

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Thanks, Russ; this is most helpful! –  David Galvin Feb 25 '12 at 3:58
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