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notes-100325 - MATH 682 1 Notes Combinatorics and Graph...

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MATH 682 Notes Combinatorics and Graph Theory II 1 Minimal Examples and Extremal Problems Minimal and extremal problems are really variations on the same question: what is the largest or smallest graph you can find which either avoids or satisfies some graph property? “Size” in these contexts is usually “number of vertices” but in some contexts is “number of edges” on a graph with a predetermined number of vertices. 1.1 Minimal examples If we are trying to find the smallest graph satisfying a certain property, it’s generally called a minimal example . For most properties, the minimal examples are pretty dull: whether we are minimizing on number of vertices or number of dges among a fixed set of vertices, most of our graph paremeters have simple minimal examples. The smallest graph of diameter d is a path on d + 1 vertices; the smallest graph of chromatic number k or clique number k is a connected graph on k vertices, and the smallest graph of independence number k is k non-adjacent vertices. These are self-evident and not terribly interesting. The one parameter we have seen so far with an interesting and not too difficult minimal example is the parameter of connectivity. Finding the fewest number of vertices on which we may connect a graph of connectivity k is not too interesting: once again, a complete graph K k +1 is best, but if we fix a number of vertices n k + 1, we might ask how many edges a graph on n vertices must have in order to have connectivity k ? It is easy to find a necessary number of edges. Proposition 1. If | G | = n and κ ( G ) = k , then k G k ≥ d nk 2 e . Proof. It has been seen previously, when first introducing the concept of connectivity, that κ ( G ) δ ( G ), so δ ( G ) k . Then 2 k G k = X v V ( G ) d ( v ) X v V ( G ) δ ( G ) = | G | δ ( G ) nk and thus k G k ≥ nk 2 , and since k G k must be an integer, we may take the ceiling of this lower bound. This almost-trivial bound will in fact completely satisfy our purposes, because we can show that specific graphs achieve it. Definition 1. The Harary graph H n,k is a graph on the n vertices { v 1 , v 2 , . . . , v n } defined by the following construction: If k is even, then each vertex v i is adjacent to v i ± 1 , v i ± 2 ,. . . , v i ± k 2 , where the indices are subjected to the wraparound convention that v i v i + n (e.g. v n +3 represents v 3 ). If k is odd and n is even, then H n,k is H n,k - 1 with additional adjacencies between each v i and v i + n 2 for each i . If k and n are both odd, then H n,k is H n,k - 1 with additional adjacencies { v 1 , v 1+ n - 1 2 } , { v 1 , v 1+ n +1 2 } , { v 2 , v 2+ n +1 2 } , { v 3 , v 3+ n +1 2 } , · · · , { v n - 1 2 , v n } Page 1 of 6 March 25, 2010
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MATH 682 Notes Combinatorics and Graph Theory II Below, the graphs H 9 , 4 , H 8 , 3 , and H 9 , 5 are shown as examples of these three classes.
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