This is the sixth part of the graph theory notes. We take a look at Hamiltonian Graphs. We omit a tactics section since the questions related to Hamiltonian Graphs are diverse each requiring some approach unique in its own way, please see the #Questions section for further references. The book followed throughout is: Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings.

This is the fifth part of the graph theory notes. We take a look at Euler’s Problem and its resolution in this post. The book followed throughout is: Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings. Regarding the Königsberg Problem The entirety of the Königsberg problem solved by Euler is covered at various places; Please refer to here - for a audiovisual treatment or here - for a full report on it.

This is the fourth part of the graph theory notes. We take a look at Trees. The book followed throughout is: Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings. Trees Before introducing trees, here are some essential terminologies: Let $G$ be a graph and its edges are $e_1,e_2,e_3,e_4,\ldots,e_m$ If edge $e_i = (v_{i-1},v_i)$ for $i = 1,2,3, \ldots, m$ exists and $v_{i-1}$ is adjacent to $v_i$ then we call the sequence a chain from $v_0$ and $v_m$.

This is the third part of the graph theory notes. We take a look at $k$-partite graphs and Turán’s Theorem. We omit the Tactics section since most problems can be handled by tactics mentioned earlier combined with the powerful results mentioned below. However, it might be worthwhile to see the connection of Inequalities (like - AM-GM, Cauchy-Schwartz etc) to problems related with finding bounds of edges. The book followed throughout is: Graph Theory with Applications by Bondy J.

This is the second part of the graph theory notes. We take a look at Degree of a Vertex and the so-called Hand-Shaking Lemma in this post. The book followed throughout is: Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings. Degree of a Vertex The degree of a vertex $v$ in a graph $G$, denoted by $d_G(v)$ (or simply, $d(v)$ if the context of graph is clear), is the number of edges incident to $v$ of $G$.