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**Graph** - In MA010 graph is a pair $G = (V, E)$ s.t. $E \subseteq {V \choose 2}$.

**Complete graph 

**Subgraph** - $H$ is subgraph of $G$ iff $V(H) \subseteq V(G)$ and $E(H) \subseteq V(H) \cap {V(G) \choose 2}$.

**Subdivision** - $H$ is a subdivision of $G$ iff it can be obtained by edge $subdivision$ meaning adding vertex $w$ and replacing $uv$ by $uw$ and $wv$ (obtained by replacing subset of edges by paths).

**Thomassen's Theorem**: For every planar graph $G$, $\chi_l(G) \le 5$.

**Voist's Theorem**: There exists a planar graph $G$ s.t. $\chi_l(G) > 4$.

**Voigt's Theorem**: There exists a planar graph $G$ s.t. $\chi_l(G) > 4$.

**Fact 2**: If $G$ is a chordal graph, then $\chi(G) = \omega(G)$.

**Fact** - A flow $f$ is optimal iff there is no _augmenting path_. 

**Ford–Fulkerson** - greedy algorithm that computes the maximum flow in a flow network using augmenting paths.

**Fact** - size of any matching $\le$ size of any vertex cover

**Kőnig's Theorem** - In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.