notes-100415

# notes-100415 - MATH 682 Notes Combinatorics and Graph...

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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Advanced Chromatic Properties 1.1 Color-Criticality When we were proving particular results in chromatic number, such as the five-color theorem, we frequently assumed we were looking at a minimal example of some species of graph. Very often, the minimal examples of a problem have exploitable structures we can make better use of. Such graphs are called critical : Definition 1. A graph G without isolated vertices is color-critical if χ ( G- e ) < χ ( G ) for any e ∈ E ( G ). If χ ( G ) = k , this property may be further described as k-criticality . We know a few color-critical graphs off the top of our heads: K n will be n-critical, since removal of any edge allows coloring with n- 1 colors, and C 2 n +1 will be 3-critical, since removing any edge admits a 2-coloring. In fact, every graph has a k-critical “core”, which in many cases is not a cycle or a clique. Proposition 1. If χ ( G ) = k , then G has a k-critical subgraph. Proof. Let us prove this by induction on k G k ; for the base case, note that k G k = 1 corresponds uniquely to a 2-critical graph. For larger G , one of two things is true: either G is k-critical, in which case it is its own k- critical subgraph, or there is an edge e such that χ ( G- e ) ≥ χ ( G ). Since any coloring of G- e is a proper coloring of G , we know that this nonstrict inequality is in fact an equality; that is, χ ( G- e ) = χ ( G ) = k . Then, by the inductive hypothesis, since k G- e k < k G k and χ ( G- e ) = k , G- e has a k-critical subgraph, which is in turn a k-critical subgraph of G . Since we now know k-critical graphs are all over the place, we might begin to wonder about their structure, since they are exemplars of the necessary substructures for a graph to require k colors. Proposition 2. If G is k-critical, then δ ( G ) ≥ k- 1 . Proof. Suppose χ ( G ) = k and G has a vertex v with degree less than k- 1. Let e = { u,v } for some neighbor u of v . If χ ( G- e ) = k- 1, then it must follow that in this coloring u and v are the same color (or this would be a ( k- 1)-coloring of G as well; however, since v has no more than k- 3 neighbors in G- e , there must be two colors ≤ k- 1 not represented in its neighborhood, so any proper coloring of G- e except for v allows at least two choices of color for v , so it can always be selected to be a different color than u , inducing a ( k- 1)-coloring of G , which contradicts the given chromatic number of G . Thus, χ ( G- e ) must be k , so G is not k-critical. Not only is the minimum degree at each vertex necessarily close to the chromatic number of a critical graph, but in fact, the more global concept of edge-connectivity must also be dictated by the chromatic number. We might start with a quite simple observation....
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notes-100415 - MATH 682 Notes Combinatorics and Graph...

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