Math 412 Class log

(1/20/2016) Passed out course outline and emergency information, definition of a graph, loops, multiple edges, simple graph, isomorphic simple graphs, independent set, clique, complement of a simple graph, self-complementary graphs, bipartite graphs, C_n, P_n and K_n.
(1/22/2016) Complete bipartite graphs, chromatic number, relationship between chromatic number and k-partite graphs, Adjacency matrix, Petersen graph, decompositions
(1/25/2016) definition of the Petersen graph, proof that the Petersen graph has girth 5, automorphism, vertex transitive graphs, the Petersen graph is vertex transitive. Strong principle of induction, walk, trail, path, Theorem 1.2.5
(1/27/2016) Connected vs adjacent, connection relation is an equivalence relation, components, trivial components and isolated vertices, Prop 1.2.11, cut-vertices, cut-edges, induced graph, 1.2.14
(1/29/2016) 1.2.15, 1.2.18 (Konig), 1.2.23 maximal vs maximum paths
(2/1/2016) min degree, max degree, 1.2.25, Eulerian, Eulerian trail, Sketched 1.2.26, 1.2.27, 1.2.28, and 1.2.29.
(2/3/2016) degree sum formula, every graph has an even number of vertices of odd degree, the hypercube, k-regular, bipartite implies parts of same size, min edges over n-vertex connected simple graphs, min degree floor of n/2 implies connected
(2/5/2016) every loopless graph has a bipartite subgraph with at least e(G)/2 edges, Mantel's Theorem, intro to graphic sequences
(2/8/2016) Havel, Hakimi, intro to directed graphs, adjacency matrix, incidence matrix, weakly/strongly connected digraphs, strong components, min outdegree or indegree at least 1 implies a cycle, characterization of Eulerian digraphs, intro to de Bruijn cycles
(2/10/2016) de Bruijn cycles, the graph D_n, Theorem 1.4.26, Orientation of a graph, Oriented graph, tournament, clear winner, king, 1.4.30
(2/12/2016) Trees, lemma 2.1.3, Theorem 2.1.4 (characterization of trees), Corollary 2.1.5,
(2/14/2016) Propositions 2.1.6, 2.1.7, 2.1.8, distance, diameter, eccentricity, 2.1.11,
(2/17/2016) center, 2.1.13 (Jordan), Wiener Index 2.1.14, 2.1.15
(2/19/2016) 2.1.16, Prufer code, Theorem 2.2.3,
(2/22/2016) Corollary 2.2.4, contraction of an edge, Prop 2.2.8, Matrix tree theorem (2.2.12) (intro)
(2/24/2016) Matrix tree theorem (2.2.12) proof
(2/26/2016) Grapeful labelings, Conjecture 2.2.13 and 2.1.15, Theorem 2.1.16, Minimum spanning trees, Kruskal's Algorithm
(2/29/2016) Shortest paths, Dijkstra's Algorithm with proof
(3/2/2016) Matchings, M-alternating paths, symmetric difference, Lemma 3.1.9, Theorem 3.1.10
(3/9/2016) Tutte's Theorem
(3/11/2016) join of two graphs, Berge-Tutte Formula, 3.3.9 (Peterson's Theorem), Intro to connectivity
(3/14/2016) Example 4.1.3, Theorem 4.1.5, Edge connectivity, edge cut, 4.1.8
(3/16/2016) Theorem 4.1.9, Theorem 4.1.11, Proposition 4.1.15, Proposition 4.1.19
(3/28/2016) Menger's Theorem (4.2.17)
(3/30/2016) Theorem 4.2.19, line graphs, Lemma 4.2.20, Global Versions of Menger's Theorem 4.2.21, Lemma 4.2.3 (Expansion Lemma)
(4/1/2016) Fan Lemma (Lemma 4.2.23), Theorem 4.2.4, Intro to flows, feasible flows, f^+, f^-, val(f)
(4/6/2016) f-augmenting path, Lemma 4.3.5, Source/Sink cut, Lemma 4.3.7, Lemma 4.2.8 (Weak Duality), Dual problem, labeling algorithm, max flow=min cut (4.3.11)
(4/8/2016) k-coloring, color class, k-colorable, k-critical, χ(G), ω(G), 5.1.7 (χ(G) >= ω(G) and χ(G) >= n(G)/α(G)), Cartesian product (5.1.11), Greedy Coloring, 5.1.13 χ(G) <= Δ(G) + 1, 5.1.14, 5.1.18 if G is k-critica, then δ(G) <= k - 1, 5.1.19,
(4/11/2016) Brooks' Theorem 5.1.22 Mycielski's construction, 5.2.3
(4/13/2016) Turán's Theorem, 5.2.8 and 5.2.9, Chromatic polynomial, 5.3.3 (If T is a tree, then χ(T; k) = k(k-1)^n),
(4/18/2016) 5.3.6 (Chromatic recurrence), Planar graphs, plane graphs, Drawings, 6.1.2 (K_5 and K_{3,3} are not planar), Dual graph, 6.1.3 (sum of face length equals two times the number of edges),
(4/20/2016) 6.1.21 Euler's Formula, 6.1.23 (G planar and n(G) >= 3 implies e(G) <= 3n(G) - 6)
(4/22/2016) 6.1.26 (characterization of maximal planar graphs) Regular polyhedra, subdivision, Kuratowski's Theorem, Every simple planar graph is 6-colorable
(4/25/2016) 6.3.1. (5-color theorem) Every simple graph is 5-colorable, Edge coloring, k-edge coloring, k-edge colorable, proper edge coloring, edge chromatic number, chromatic index, Vizing's Theorem (If G simple, then χ'(G) <= Δ(G) + 1), The chromatic index of the Petersen graph is 4.
(4/27/2016) Full Vizing's Theorem (χ'(G) <= Δ(G) + μ(G)) 1-factorization, 1-factorable, If G is a k-regular bigraph, then G is 1-factorable, (7.1.7) If G is bipartite, then χ'(G) = Δ(G). Hamlitonian cycles, Hamilonian graphs, K_{m,n} is hamiltonian iff m=n >= 2, If G is a Hamiltonian bipartite graph, then the parts have the same size, The Petersen graph is not Hamiltonian, c(H)=number of components of H, G Hamiltonian implies c(G-S) <= |S| for every non-empty S a subset of V(G).
(4/29/2016) Dirac's Theorme (7.2.8) Chvatal-Erdos (7.2.19)
(5/2/2016) Finish Chvatal-Erdos (7.2.19) and class overview