Advanced Graph Theory: Course Outline (MAT-620)

What is a graph, Simple graphs, Graph and their Representations, Isomorphism and automorphisms, Labelled graphs, Graphs arising from other structures, Incidents Graphs, Union and Intersection Graphs ,Cartesian Product, Direct Graph, Sub-graphs and Supergraphs, spanning and induced sub-graphs, Decomposition and coverings, Edge cuts and bonds, even sub-graphs, Graph reconstruction, Walks and connection, Cut edges, Connection to diagraphs, Cycle double covers, Forests and Trees, Spanning Tree, Calay’s formula, Fundamental Cycles and Bonds, Co-tree, Trees and Distance. Applications of Tree, Cut vertices, Separations and Blocks, Ear Decompositions, Strong Orientations, Directed Ear Decompositions, Even Cycles Decompositions, Vertex Connectivity, Fan Lemma, Edge Connectivity, Three-connected graphs, Sub- modularity, Determining, Chordal graphs, Simplicial vertices, Plane and Planar graphs, Duality, Euler’s formula, Bridges, Kuratowski’s theorem, Chromatic numbers, Critical graphs, Girth and chromatic number, Perfect graphs, List colorings’, The adjacency polynomial, Chromatic polynomial. Applications of vertex colorings to various problems. Edge colouring number, Vizing’s theorem, Snarks, Covering by perfect matching, List edge coloring, Applications of edge colorings. Applications to various selected problems. Hamiltonian and non-Hamiltonian graphs, Non-Hamiltonian planar.

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