Mycielski’s Construction

v1 u1 v2 u2 v5 v2 u2 u5 w w u4 u3 v1 u1 v4 v3 Mycielski’s Construction • Mycielski’s Construction: From a simple graph G, Mycielski’s Construction produces a simple graph G’ containing G. Beginning with G having vertex set {v1, v2, …,vn}, add vertices U={u1, u2, …,un} and one more vertex w. Add edges to make ui adjacent to all of NG(vi), and finally let NG’(w)=U. v2 v1

Theorem 5.2.3 From a k-chromatic triangle-free graph G, Mycielski’s construction produces a k+1-chromatic triangle-free graph G’. Proof. 1. Let V(G)={v1, v2, …,vn}, and let G’ be the graph produced from it by Mycielski’s construction. Let u1, u2, …,un be the copies of v1, v2, …,vn, with the additional vertex w. Let U={u1, u2, …,un}.

Theorem 5.2.3 2. Suppose G’ has a triangle. 3. The triangle contains at least one node in U, say ui, since G is triangle-free. 4. Since U is an independent set in G, the other vertices of the triangle belong to V(G), say vj, vk. 5. vj, vk are neighbors of vi. 6. There are a triangle vi, vj, vk in G, which is a contradiction. v1 u1 v5 v2 u2 u5 w u4 u3 v4 v3

Theorem 5.2.3 7. A proper k-coloring f of G extends to a proper k+1-coloring of G’ by setting f(ui)=f(vi) and f(w)=k+1  (G’)<= (G)+1. v1 u1 v5 v2 u2 u5 w u4 u3 v4 v3

Theorem 5.2.3 8. Suppose G’ has a proper k-coloring g. 9. By changing the names of colors, we may assume g(w)=k. This restricts g to {1, 2, …, k-1} on U. 10. For each vion which g uses color k, we change the color of vi to g(ui). 11. The modified coloring g’ of V(G) is a proper k-1-coloring of G, which is a contradiction. v1 v1 u1 u1 v5 v5 v2 v2 u2 u5 w u4 u3 u3 v4 v3 v4 v3

Proposition 5.2.5 Every k-chromatic graph with n vertices has at least k*(k-1)/2 edges. Proof. 1.At least one edge with endpoints of colors i and j for each pair i, j of colors. Otherwise, colors i and j could be combined into a single color class and use fewer colors. 2. It implies at least k*(k-1)/2 edges in k-chromatic graph with n vertices.

Complete Multipartite Graph Complete Multipartite Graph: A complete multipartite graph is a simple graph G whose vertices can be partitioned into sets so that (u,v)E(G) if and only if u and v belongs to different sets of the partition. Equivalently, every component of G is a complete graph. When k>=2, we write Kn1n2nk for the completek-partite graph with partite sets of size n1, …, nk and complement Kn1+ …+Knk. K1,3,4

Turan Graph Turan Graph: The Turan graph Tn,r is the complete r-partite graph with n vertices whose partite sets differ in size by at most 1. That is, all partite sets have size n/r or n/r. T8,3

Lose j neighbors Gain i-1 neighbors Lemma 5.2.8 Among simple r-partite graphs with n vertices, the Turan graph is the unique graph with the most edges. Proof. 1. We need only consider complete r-partite graphs. 2. Given a complete r-partite graph with partite sets differing by more than 1 in size, we move a vertex v from the largest size (size i) to the smallest class (size j). 3. It is easy to verify the number of edges increases.

Theorem 5.2.9 Among the n-vertex simple graphs with no r+1-clique, Tn,r has the maximum number of edges. Proof. 1. Tn,r has no r+1-clique. 2. It suffices to show the maximum is achieved by an r-partite graph by Lemma 5.2.8. 3. It is proved that if G has no r+1-clique, then there is an r-partite graph H with the same vertex set as G and at least as many edges by induction on r. 4. When r=1, G and H have no edges.

Theorem 5.2.9 5. Consider r>1. Let G be an n-vertex graph with no r+1-clique, and let xV(G) be a vertex of degree k=(G). 6. Let G’ be the subgraph of G induced by the neighbors of x (N(x)) and let S=V(G)-N(x). 7. G’ has no r-clique because x is adjacent to every vertex in G’ and G has no r+1-clique. 8. By induction hypothesis, there is a r-1-partite graph H’ with vertex set V(G’) such that e(H’)>=e(G’). x x H’ S S-x G’ G H

Theorem 5.2.9 9. Let H be the graph formed from H’ by joining all of V(G’) to all of S. 10. H is r-partite because S is an independent set in H. 11. e(G)<=e(G’)+vSdG(v)<=e(G’)+k(n-k). 12. e(H)=e(H’)+k(n-k). 13. e(G)<=e(H) because e(G’)<=e(H’). x x k(n-k) edges H’ S S-x G’ G H

Lemma 5.2.15 Let G be a graph with (G)>k, and let X,Y be a partition of V(G). If G[X] and G[Y] are k-colorable, then the edge cut [X,Y] has at least k edges. Proof. 1. Let X1,…,Xk and Y1,…,Yk be the partitions of X and Y formed by the color class in proper k-colorings of G[X] and G[Y]. 2. If there is no edge between Xi and Yj, then XiYj is an independent set in G. In this case, Xi and Yj can have the same color.

Lemma 5.2.15 3. Form a bipartite graph H with vertices X1,…,Xk and Y1,…,Yk, putting XiYjE(H) if in G there is no edge between the set Xi and the set Yj. 4. It suffices to show if |[X,Y]|<k, then H has a perfect matching. (If H has a perfect matching, G has a proper k-coloring which constitutes a contradiction.)

Lemma 5.2.15 5. Suppose |[X,Y]|<k. 6. H has more than k(k-1) edges. 7. E(H) cannot be covered by k-1 vertices because m vertices can cover at most km edges in a subgraph of Kk,k. 8. The minimum size of a vertex cover in H is at least k. 9. The maximum size of a matching in H is at least k by Theorem 3.1.16. 10. H has a perfect matching.

Theorem 5.2.16 Every k-critical graph is k-1-edge-connected. Proof. 1. Let G be a k-critical graph, and let [X,Y] be a minimum edge cut. 2. G is k-critical, G[X] and G[Y] are k-1-colorable. 3. |[X,Y]|>=k-1.

Mycielski’s Construction