Math 412 Class log

(1/22/2014) Passed out course outline and emergency information, definitions, Konigsberg bridge problem, job assignment problem
(1/24/2014) definitions: walk, trail and path, Lemma 1 (Lemma 1.2.5 in the book) "Every u,v-walk contains a u,v-path", defined connected, connected graph and components, discussion about characterizing a collection of graphs, started proof of Theorem 1 (Theorem 1.2.18 in the book) "A graph is bipartite if and only it contains no odd cycles" with proof of Lemma 2 (Lemma 1.2.15 in the book) "Every odd closed walk contains an odd cycle" (will review this proof in next lecture), passed out homework 1.
(1/27/2014) Finish proof of Theorem 3 (Theorem 1.2.18 in the book) "A graph is bipartite if and only it contains no odd cycles", defined complete graphs and complete bipartite graphs, defined the adjacency and incidence matrices of a graph (will provide examples on Wednesday)
(1/29/2014) Adjacency and incidence matrices of a graphs with examples, defined isomorphism, degree, and regular graphs, Proved Lemma 4 "If the minimum degree of G is at least 2, then G contains a cycle." (Lemma 1.2.25 in the book), defined decomposition, and even graphs, proved Proposition 5 (Proposition 1.2.27 in the book) "Every even graph decomposes into cycles"
(1/31/2014) Proved Proposition 6 "Every simple graph G with minimum degree k has a path of length k and when k is greater than 2 G contains a cycle of length at least k + 1" (Proposition 1.2.28 in the book), Proved Theorem 7 "A graph is Eulerian if and only if it is even and has at most one non-trivial component" (Theorem 1.2.26 in the book), Proved Proposition 1.3.3 "Degree-Sum formula" (Proposition 1.3.3 in the book) passed out homework 2.
(2/3/2014) Corollaries to the degree-sum formula, degree sequences and graphic sequences, Proved Theorem 8 "Havel-Hakimi" (Theorem 1.3.31) stated Mantel's Theorem (1.3.23)
(2/5/2014) Brief discussion of extremal graph theory, began proof of Turán's theorem (5.2.9)
(2/7/2014) Finished Turán's theorem (5.2.9), initial definitions related to directed graphs. passed out homework 3.
(2/10/2014) Hypercubes, More definition related to directed graphs, degree sum formula for directed graphs (prop 1.4.18), sketch of proof of 1.4.23 and 1.4.24, discussed de Bruijn cycles and de Bruijn graphs and stated Theorem 1.4.26.
(2/12/2014) Proved Theorem 1.4.26 on de Bruijn cycles. Oriented graphs, Tournaments, Proved Proposition 1.4.30 on kings in Tournaments
(2/14/2014) Forests, Trees, Lemma 2.1.3 on leaves of trees, cut-vertices and cut-edges, proved Theorem that characterizes cut-edges, (theorem 1.2.14), Stated and began proving Theorem 2.1.4 on characterizing trees, Quiz 1, homework 4 assigned.
(2/17/2014) Short discussion on Quiz 1 (Eulerian trials vs Eulerian circuits Definition 1.2.24), Finished proof of Theorem on characterizing trees (Theorem 2.1.4), Proved Corollary 2.1.5. Distance, diameter, eccentricity and radius, definition of the center of a graph, stated Jordan's Theorem on center of trees (Theorem 2.1.13),
(2/19/2014) Proved Jordan's Theorem on center of trees (Theorem 2.1.13), Discussion on counting of the number of spanning trees, introduced Prüfer codes, stated Cayley's formula (Theorem 2.2.3)
(2/21/2014) finished proof of Cayley's formula (Theorem 2.2.3), discussed edge deletion and contraction and a introduced a recurrence for the number of spanning trees Proposition (2.2.8) homework 5 assigned.
(2/24/2014) finished proof of Proposition (Theorem 2.2.8), stated the Matrix Tree theorem (Theorem 2.2.12)
(2/26/2014) Test 1
(2/28/2014) Finished matrix tree theorem (Theorem 2.2.12), introduced the minimum spanning tree problem
(3/03/2014) Proof that Kruskal's Algorithm produces a minimum spanning tree (Theorem 2.3.3), Introduced matchings, alternating and augmented paths Started (Lemma 3.1.9) "Every component of the symmetric difference of two matchings is a path or even cycle",
(3/05/2014) (Lemma 3.1.9) "Every component of the symmetric difference of two matchings is a path or even cycle", (Theorem 3.1.10) "A matching M in a graph G is maximum if and only if M has no M-augmenting paths" Started Hall's Theorem (Theorem 3.1.11)
(3/07/2014) Finished Hall's Theorem (Theorem 3.1.11), Marriage theorem (Theorem 3.1.13), Started stable marriage theorem - Gale-Shapley (Theorem 3.2.18) homework 6 assigned.
(3/10/2014) finished stable marriage theorem (Theorem 3.2.18), Introduced vertex cover, König, Egerváry "If G is bipartite, then the size of a maximum matching is equal to the size of a minimum vertex cover, i.e. α'(G) = β(G)" Introduced edge cover, (Lemma 3.1.21) "α(G) + β(G) = n(G)"
(3/12/2014) Finished Lemma 3.1.21 "α(G) + β(G) = n(G)", proved Theorem 3.1.22 "If G has no isolated vertices, then α'(G) + β'(G) = n(G)", and corollary 3.1.24 "If G is bipartite without isolated vertices, then α(G) = β'(G)", Quiz 2
(3/14/2014) Introduction to matching in general graphs, Obs "only if part of Tutte's Theorem", observation "n(G) and o(G-S) - |S| have the same parity", discussion on an edge-maximal counterexample, beginning of Tutte's Theorem (3.3.3)
(3/17/2014) Finished Tutte's Theorem (3.3.3), Petersen's Theorem (3.3.8), Stated Berge-Tutte formula (3.3.7),
(3/19/2014) Finished Berge-Tutte formula (3.3.7) and Theorem 3.3.9 "Every 2k-regular graph has a 2-factor", short discussion of Julius Petersen, factor decompositions of regular graphs and the Petersen graph
(3/21/2014) Intro to connectivity, separating set, k-connected, connectivity of a graph, disconnecting set, edge connectivity, edge cut, equivalence of a minimal disconnecting and an edge cut, if G is a simple graph κ(G) <= κ'(G) <= δ(G) (Theorem 4.1.9)
(3/31/2014) if G is 3-regular κ(G) = κ'(G) (Theorem 4.1.11) Intro to Menger's Theorem Started Menger's Theorem
(4/2/2014) Test 2
(4/4/2014) Finished Menger's Theorem, Proved a variant of Menger's Theorem (Theorem 4.2.17),
(4/7/2014) Introduced the line graph, Proved edge version of Menger's Theorem (Theorem 4.2.19), Lemma 4.2.20, and Global version of Menger's Theorem (Theorem 4.2.21)
(4/9/2014) Passed back test 2, Fan-lemma, Discussed homework problem, Intro to flows, Value of a flow, Augmenting path, tolerance of path, Lemma 4.3.5
(4/11/2014) definition of a source/sink cut and the capacity of a cut, flow out of and into of a cut, Lemma 4.3.7, Corollary 4.3.8 (Weak Duality), Algorithm 4.3.9 and examples.
(4/14/2014) Theorem 4.3.11 (Max-flow min-cut Theorem - Ford - Fulkerson) Theorem 4.2.8 (Ear-Decomposition Theorem)
(4/16/2014) Lemma 6.2.9 (Every 3-connected graph G with at most 5 vertices has an edge e such that G.e is 3-connected). Planar graphs, plane graphs, crossings, polygonal arcs, polygons
(4/18/2014) Planar Graphs: Jordan Curve Theorem, Euler's Formula, Dual Graph, Proposition 6.1.13 (if G is a plane graph 2e(G) is the sum of the lengths of the faces of G), Definitions: open, open ball, regions, faces, outerface, outerplanar, quiz 3
(4/21/2014) Planar graphs: Theorem 6.1.23, Proposition 6.1.26, Statement of Theorem 6.2.2 (Kuratowski's Theorem), started proof of Theorem 6.2.5 Definitions: Maximal planar graph, minimal nonplanar graph, triangulation, subdivision, branch vertices, S-lobe
(4/23/2014) Continued Kuratowski's Theorem - Finished Lemma 6.2.5, Lemma 6.2.6 and Lemma 6.2.7
(4/25/2014) Lemma 6.2.10, Started 6.2.11, Quiz 4
(4/28/2014) Finished Kuratowki's Theorem - Theorem 6.2.11, introduction to coloring, Proposition 5.1.7 χ(G) >= n(G)/α(G), started Proposition 5.1.13 χ(G) <= Δ(G) + 1 Definition: ω(G) (clique number), Example: ω(G) < χ(G)
(4/30/2014) Test 3
(5/2/2014) 6-color theorem, 5-color theorem (Theorem 6.3.1), short discussion on the 4-color theorem, Hamiltonian cycles and Dirac's Theorem (7.2.8)
(5/5/2014) Test 4
(5/7/2014) Mycielski's construction (5.2.1) and relelated Theorem (5.2.3)