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Mycielski’s Construction

Mycielski’s Construction.

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Mycielski’s Construction

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  1. 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.

  2. 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 w the additional vertex. Let U={u1, u2, …,un}.

  3. Theorem 5.2.3 (2/4) 2. G’ is triangle-free. Suppose G’ has a triangle.  The triangle contains at least one node in U, say ui, since G is triangle-free.  Since U is an independent set in G, the other vertices of the triangle belong to V(G), say vj, vk.  vj, vk are neighbors of vi.  There are a triangle vi, vj, vk in G.  It is a contradiction.

  4. Theorem 5.2.3 (3/4) 3. 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. 4. The equality can be proved by showing (G)< (G’). To prove this we consider any proper coloring of G’ and obtain from it a proper coloring of G using fewer colors.

  5. Theorem 5.2.3 (3/4) 5. Let g be a proper k-coloring of G’. By changing the names of colors, we may assume g(w)=k. This restricts g to {1, 2, …, k-1} on U. 6. On V(G), it may use all k colors. Let A be the set of vertices in G on which g uses color k. It suffices to change the colors used on A to obtain a proper k-1-coloring of G. 7. For each viA, we change the color of vi to g(ui). 8. We need to prove the modified coloring g’ of V(G) is a proper k-1-coloring of G.

  6. Theorem 5.2.3 (4/4) 9. Let vi, vj be two adjacent vertices in G. 10. vi, vj have different colors under g. We need to prove vi, vj have different colors under g’. 11. All vertices of A have color k under g.  No two vertices of A are adjacent.  At most one of vi, vj is in A. 12. Case 1: vi, vjA. The colors of vi, vj are not changed. vi, vj have different colors under g’. 13. Case 2: viA and vjA. By construction, (ui,vj)E(G).  ui, vj have different colors under g.  vi, vj have different colors under g’.

  7. Proposition 5.2.5 Every k-chromatic graph with n vertices has at least k*(k-1)/2 edges. Proof. 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.  At least k*(k-1)/2 edges in k-chromatic graph with n vertices.

  8. Turan 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. 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.

  9. 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. The edges not involving v are the same as before, but v gains i-1 neighbors in its old class and loses j neighbors in its new class. 4. Since i-1>j, the number of edges increases.  We maximize the number of edges only by equalizing the size as in Tn,r.

  10. 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. If we can prove that the maximum is achieved by an r-partite graph, then Lemma 5.2.8 implies that the maximum is achieved by Tn,r. 3. It suffices to prove 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.

  11. Theorem 5.2.9 4. This is proved by induction on r. 5. When r=1, G and H have no edges. 6. 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). 7. Let G’ be the subgraph of G induced by the neighbors of x. 8. x is adjacent to every vertex in G’ and G has no r+1-clique.  The graph G’ has no r-clique.  By induction hypothesis, there is a r-1-partite graph H’ with vertex set N(x) such that e(H’)<=e(G’).

  12. Theorem 5.2.9 10. Let H be the graph formed from H’ by joining all of N(x) to all of S=V(G)-N(x). 11. S is an independent set.  H is r-partite. 12. We need to prove e(H)>=e(G). 13. By construction, e(H)=e(H’)+k(n-k). 14. e(G)<=e(G’)+vSdG(v)<=e(H’)+k(n-k)<=e(H).

  13. 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. 3. We show that if |[X,Y]|<k, then we can combine color classes from G[X] and G[Y] in pairs to form a proper k-coloring of G.

  14. Lemma 5.2.15 (2/3) 4. 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. 5. Suppose that |[X,Y]|<k. Then, H has more than k(k-1) edges.

  15. Lemma 5.2.15 (3/3) 6. m vertices can cover at most km edges in a subgraph of Kk,k  E(H) cannot be covered by k-1 vertices.  The minimum size of a vertex cover in H is at least k.  The maximum size of a matching in H is at least k by Theorem 3.1.16.  H has a perfect matching M. 7. In G, we give color i to all of Xi and all of Yj to which it is matched by M. 8. There are no edges joining Xi and Yj, doing this for all i produces a proper k-coloring of G.  It contradicts to the hypothesis that (G)>k.  |[X,Y]|>=k.

  16. 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.

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