- #[[MA284 - Discrete Mathematics]] - **Previous Topic:** [[Introduction to Graph Theory]] - **Next Topic:** [[Convex Polyhedra]] - **Relevant Slides:** ![MA284-Week08.pdf](../assets/MA284-Week08_1666785726176_0.pdf) - - # Definitions - What is a **walk**? #card card-last-interval:: 2.8 card-repeats:: 2 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-26T07:16:18.888Z card-last-reviewed:: 2022-11-23T12:16:18.890Z card-last-score:: 5 - A **walk** is a sequence of vertices such that consecutive vertices are adjacent. - What is a **trail**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:14:59.935Z card-last-score:: 1 - A **trail** is a walk in which no edge is repeated. - What is a **path**? #card card-last-interval:: 0.9 card-repeats:: 2 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-15T17:22:40.523Z card-last-reviewed:: 2022-11-14T20:22:40.524Z card-last-score:: 3 - A **path** is a trail in which no vertex is repeated, except possibly the first & last. - The path on $n$ vertices is denoted $P_n$. - ## Example - {{renderer :mermaid_uiukfrr}} - ```mermaid flowchart LR B((B)) --- A((A)) B --- C((C)) C --- E((E)) E --- D((D)) B --- E --- A ``` - $(a,b,c,e,d)$ is a **walk**. - So too is $(a,b,e,a,b,c)$ (not a trail or a path). - - What is the **length** of a path? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:22:48.413Z card-last-score:: 1 - The **length** of a path is the number of edges in the sequence. - ## Cycles & Circuits - What is a **cycle**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:50:16.908Z card-last-score:: 1 - A **cycle** is a path that begins & ends at the same vertex, but no other vertex is repeated. - A cycle on $n$ vertices is denoted $C_n$. - What is a **circuit**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-22T00:00:00.000Z card-last-reviewed:: 2022-11-21T13:05:28.290Z card-last-score:: 1 - A **circuit** is a path that begins & ends at the same vertex, and no edge is repeated. - What does it mean if a graph is **connected**? #card card-last-interval:: 2.8 card-repeats:: 1 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-17T11:16:09.571Z card-last-reviewed:: 2022-11-14T16:16:09.572Z card-last-score:: 5 - A graph is **connected** if there is a path between every pair of vertices. - What is the **degree** of a vertex? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.7 card-next-schedule:: 2022-11-25T13:10:12.122Z card-last-reviewed:: 2022-11-21T13:10:12.123Z card-last-score:: 5 - The **degree** of a vertex is the number of edges emanating from it. - If $v$ is a vertex, we denote its degree as $d(v)$. - ## Handshaking Lemma - If we know the degree of every vertex in the graph, then we know the number of edges. This is the **Handshaking Lemma**. - What is the **Handshaking Lemma**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:54:11.744Z card-last-score:: 1 - In any graph, the sum of the degrees of vertices in the graph, is always twice the number of edges. - $$\sum_{v \in V} d(v) = 2|E|$$ - # Types of Graphs - What is a **Complete** Graph? #card card-last-interval:: 0.9 card-repeats:: 2 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-15T17:23:00.989Z card-last-reviewed:: 2022-11-14T20:23:00.990Z card-last-score:: 3 - A graph is **complete** if every pair of vertices is adjacent. - This family of graphs is very important. - Complete graphs are denoted $K_n$ - the complete graph on $n$ vertices. - What is a **Bipartite Graph**? #card card-last-interval:: 8.35 card-repeats:: 3 card-ease-factor:: 2.46 card-next-schedule:: 2022-11-29T21:09:36.899Z card-last-reviewed:: 2022-11-21T13:09:36.899Z card-last-score:: 5 - A graph is **bipartite** if it is possible to partition the vertex set, $V$, into two disjoint sets, $V_1$ & $V_2$, such that there are no edges between any two vertices in the same set. - What is a **Complete Bipartite** graph? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:51:45.394Z card-last-score:: 1 - If a bipartite graph is such that *every* vertex in $V_1$ is connected to *every* vertex in $V_2$ (and vice versa), the graph is a **complete bipartite graph**. - If $|V_1| = m$ and $|V_2| = n$, we denote it $K_{m,n}$. - What is a **subgraph**? #card card-last-interval:: 3.05 card-repeats:: 2 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-17T21:23:07.436Z card-last-reviewed:: 2022-11-14T20:23:07.436Z card-last-score:: 5 - We say that $G_1 = (V_1, E_1)$ is a **subgraph** of $G_2 = (V_2, E_2)$ provided $V_1 \subset V_2$ and $E_1 \subset E_2$. - What is an **induced subgraph**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-23T00:00:00.000Z card-last-reviewed:: 2022-11-22T13:38:04.242Z card-last-score:: 1 - We say that $G_1(V_1, E_1)$ is an **induced subgraph** of $G_2 = (V_2, E_2)$ provided that $V_1 \subset V_2$ and $E_2$ contains **all** edges of $E_1$ which join edges in $V_1$. - - # Planar Graphs - What is a **planar graph**? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:19:45.615Z card-last-score:: 1 - If you can sketch a graph such that none of its edges cross, it is a **planar graph**. - What is a **face**? #card card-last-interval:: 0.98 card-repeats:: 1 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-15T15:22:52.013Z card-last-reviewed:: 2022-11-14T16:22:52.013Z card-last-score:: 3 - When a planar graph is drawn without edges crossing, the edges & vertices of the graph divide the plane into regions called **faces**. - The number of faces does not change no matter how you draw the graph, as long as no edges cross. - - ## Example - The graph $K_{2,3}$ is **planar**. background-color:: red - ![image.png](../assets/image_1666951300835_0.png) - [[draws/2022-10-28-11-04-05.excalidraw]] - The planar representation $K_{2,3}$ has **3 faces** (the "outside" region counts as a face). - Give a planar representation of $K_4$, and count how many faces it has. background-color:: red - [[draws/2022-10-28-11-22-12.excalidraw]] - Why "face"? background-color:: red - [[draws/2022-10-28-11-25-20.excalidraw]] - # Euler's Formula for Planar Graphs #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:18:07.821Z card-last-score:: 1 - For any ^^(connected) planar graph^^ with $v$ vertices, $e$ edges, and $f$ faces, we have: - $$v - e + f = 2$$ - ## Outline of Proof - Start with $P_2$. - Here, $v=2$, $e = 1$, $f=1$. So $v-e+f=2$. - Any other graph can be made by adding vertices & edges (or just edges) to $P_2$. - Suppose $v-e+f=2$ for a graph. - If we add a new edge *with* a new vertex, then no new face is created, so $v-e+f$ does not change. - If we add a new edge *without* a new vertex, then $f$ will increase by 1, so again, $v-e+f$ does not change. - ## Example - Is it possible for a connected planar graph to have 5 vertices, 7 edges, and 3 faces? Explain. background-color:: red - No. Euler's formula tells us that $v-e+f=2$. - Here, $v=5$, $e=7$, $f=3$, so $v-e+f=1$. - Any such graph is **not planar**. - -