🚧 Setup for 4.9

chapter_4/exercises.md

@@ -8823,3 +8823,143 @@ Omitted.
     _multiplicative identity_.)
 
 Omitted.
+
+---
+
+**Exercise Set 4.9**
+
+Page 265
+
+In 1 and 2 find the degree of each vertex and the total degree of the graph.
+Check that the number of edges equals one-half of the total degree.
+
+1. See page 265.
+
+2. See page 265.
+
+3. A graph has vertices of degrees 0, 2, 2, 3, and 9. How many edges does the
+   graph have?
+
+4. A graph has vertices of degrees 1, 1, 4, 4, and 6. How many edges does the
+   graph have?
+
+In each of 5-13 either draw a graph with the specified properties or explain why
+no such graph exists.
+
+5. Graph with five vertices of degrees 1, 2, 3, 3, and 5.
+
+6. Graph of four vertices of degrees 1, 2, 3, and 3.
+
+7. Graph with four vertices of degrees 1, 1, 1, and 4.
+
+8. Graph with four vertices of degrees 1, 2, 3, and 4.
+
+9. Simple graph with four vertices of degrees 1, 2, 3, and 4.
+
+10. Simple graph with five vertices of degrees 2, 3, 3, 3, and 5.
+
+11. Simple graph with five vertices of degrees 1, 1, 1, 2, and 3.
+
+12. Simple graph with six edges and all vertices of degree 3.
+
+13. Simple graph with nine edges and all vertices of degree 3.
+
+14. At a party attended by a group of people, two people knew one other person
+    before the party, and five people knew two other people before the party.
+    The rest of the people knew three other people before the party. A total of
+    15 pairs of people knew each other before the party.
+
+a. How many people attending the party knew three other people before the party?
+
+b. How many people attended the party?
+
+15. A small social network contains three people who are network friends with
+    six other people in the network, one person who is network friend with five
+    other people in the network, and five people who are network friends with
+    four other people in the network. The rest are network friends with three
+    other people in the network. The network contains 41 pairs of network
+    friends.
+
+a. How many people are network friends with three other people in the network?
+
+b. How many people are in the network?
+
+16.
+
+a. In a group of 15 people, is it possible for each person to have exactly 3
+friends? Justify your answer. (Assume that friendship is a symmetric
+relationship: If $x$ is a friend of $y$, then $y$ is a friend of $x$.)
+
+b. In a group of 4 people, is it possible for each person to have exactly 3
+friends? Justify your answer.
+
+17. In a group of 25 people, is it possible for each to shake hands with exactly
+    3 other people? Justify your answer.
+
+18. Is there a simple graph, each of whose vertices has even degree? Justify
+    your answer.
+
+19. Suppose that $G$ is a graph with $v$ vertices and $e$ edges and that the
+    degree of each vertex is at least $d_{\text{min}}$ and at most
+    $d_{\text{max}}$. Show that
+
+$$ \frac{1}{2}d_{\text{min}} \cdot v \leq e \leq \frac{1}{2}d_{\text{max}} \cdot v $$
+
+20.
+
+a. Draw $K_6$, a complete graph on six vertices.
+
+b. Use the result of Example 4.9.9 to show that the number of edges of a simple
+graph with $n$ vertices is less than or equal to $\dfrac{n(n - 1)}{2}$.
+
+21.
+
+a. In a simple graph, must every vertex have degree that is less than the number
+of vertices in the graph? Why?
+
+b. Can there be a simple graph that has four vertices all of different degrees?
+Why?
+
+c. For any integer $n \geq 5$, can there be a simple graph that has $n$ vertices
+all of different degrees? Why?
+
+22. In a group of two or more people, must there always be at least two people
+    who are acquainted with the same number of people within the group? Why?
+
+23. Recall that $K_{m, n}$ denotes a complete bipartite graph on $(m, n)$
+    vertices.
+
+a. Draw $K_{4, 2}$.
+
+b. Draw $K_{1, 3}$.
+
+c. Draw $K_{3, 4}$.
+
+d. How many vertices of $K_{m, n}$ have degree $m$? degree $n$?
+
+e. What is the total degree of $K_{m, n}$?
+
+f. Find a formula in terms of $m$ and $n$ for the number of edges of $K_{m, n}$.
+Justify your answer.
+
+24. A (general) **bipartite graph** $G$ is a simple graph whose vertex set can
+    be partitioned into two disjoint nonempty subsets $V_1$ and $V_2$ such that
+    vertices in $V_1$ may be connected to vertices in $V_2$, but no vertices in
+    $V_1$ and no vertices in $V_2$ are connected to other vertices in $V_2$. For
+    example, the bipartite graph $G$ illustrated in (i) can be redrawn as shown
+    in (ii). From the drawing in (ii), you can see that $G$ is bipartite with
+    mutually disjoint vertex sets $V_w = \{v_1, v_3, v_5\}$ and
+    $V_2 = \{v_2, v_4, v_6\}$.
+
+(i) See Page 266
+
+(ii) See Page 266
+
+Find which of the following graphs are bipartite. Redraw the bipartite graphs so
+that their bipartite nature is evident.
+
+See Page 266.
+
+25. Suppose $r$ and $s$ are any positive integers. Does there exist a graph $G$
+    with the property that $G$ has vertices of degrees $r$ and $s$ and no other
+    degrees? Explain.

chapter_4/notes.md

@@ -1110,3 +1110,135 @@ does not divide $(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \dots p) + 1$, which equals
 $N$. Hence $N$ is divisible by $q$ and $N$ is not divisible by $q$, and we have
 reached a contradiction. _[Therefore, the supposition is false and the theorem
 is true.]_
+
+---
+
+Page 258
+
+**Definition**
+
+The total degree of a graph is the sum of the degrees of all the vertices of the
+graph.
+
+---
+
+Page 259
+
+**Theorem 4.9.1 The Handshake Theorem**
+
+If $G$ is any graph, then the sum of the degrees of all the vertices of $G$
+equals twice the number of edges of $G$. Specifically, if the vertices of $G$
+are $v_1, v^2, \dots v_n$, where $n$ is a nonnegative integer, then
+
+$$ \text{the total degree of } G = \text{deg}(v_1) + \text{deg}(v_2) + \dots + \text{deg}(v_n) $$
+
+$$ = 2 \cdot (\text{the number of edges of } G) $$
+
+**Proof:**
+
+Let $G$ be a particular but arbitrarily chosen graph, and suppose that $G$ has
+$n$ vertices $v_1, v_2, \dots v_n$ and $m$ edges, where $n$ is a positive
+integer and $m$ is a nonnegative integer. We claim that each edge of $G$
+contributes $2$ to the total degree of $G$. For suppose $e$ is an arbitrarily
+chosen edge with endpoints $v_i$ and $v_j$. This edge contributes $1$ to the
+degree of $v_i$ and $1$ to the degree of $v_j$. As shown below, this is true
+even if $i = j$, because an edge that is a loop is counted twice in computing
+the degree of the vertex on which it is incident.
+
+(see Page 259)
+
+Therefore, $e$ contributes $2$ to the total degree of $G$. Since $e$ was
+arbitrarily chosen, this shows that _each_ edge of $G$ contributes $2$ to the
+total degree of $G$. Thus
+
+$$ \text{the total degree of } G = 2 \cdot (\text{the number of edges of } G) $$
+
+---
+
+Page 259
+
+**Corollary 4.9.2**
+
+The total degree of a graph is even.
+
+**Proof:** By Theorem 4.9.1 the total degree of $G$ equals $2$ times the number
+of edges of $G$, which is an integer, and so the total degree of $G$ is even.
+
+---
+
+Page 260
+
+**Proposition 4.9.3**
+
+In any graph there is an even number of vertices of odd degree.
+
+**Proof:** Suppose $G$ is any graph, and suppose $G$ has $n$ vertices of odd
+degree and $m$ vertices of even degree, where $n$ is a positive integer and $m$
+is a nonnegative integer. _[We must show that $n$ is even.]_ Let $E$ be the sum
+of the degrees of all the vertices of even degree, $O$ the sum of the degrees of
+all the vertices of odd degree, and $T$ the total degree of $G$. If
+$u_1, u_2, \dots, u_m$ are the vertices of even degree and
+$v_1, v_2, \dots, v_n$ are the vertices of odd degree, then
+
+$$ E = \text{deg}(u_1) + \text{deg}(u_2) + \dots + \text{deg}(u_m), $$
+
+$$ O = \text{deg}(v_1) + \text{deg}(v_2) + \dots + \text{deg}(v_m), \text{ and} $$
+
+$$ T = \text{deg}(u_1) + \dots + \text{deg}(u_m) + \text{deg}(v_1) + \dots + \text{deg}(v_n) = E + 0 $$
+
+Now $T$, the total degree of $G$, is an even integer by Corollary 4.9.2. Also
+$E$ is even since either $E$ is zero, which is even, or $E$ is a sum of even
+numbers. Now since
+
+$$ T = E + O $$
+
+then
+
+$$ O = T - E $$
+
+Hence $O$ is a difference of two even integers, and so $O$ is even.
+
+By assumption, $\text{deg}(v_i)$ is odd for every integer $i = 1, 2, \dots, n$.
+Thus $O$, an even integer, is a sum of the $n$ odd integers
+$\text{deg}(v_1), \text{deg}(v_2), \dots, \text{deg}(v_n)$. But if a sum of $n$
+odd integers is even, then $n$ is even. Therefore, $n$ is even _[as was to be
+shown]._
+
+---
+
+Page 262
+
+**Definition and Notation**
+
+A **simple graph** is a graph that does not have any loops or parallel edges. In
+a simple graph, an edge with endpoints $v$ and $w$ is denoted $\{v, w\}$.
+
+---
+
+Page 263
+
+**Definition**
+
+Let $n$ be a positive integer. A **complete graph on $n$ vertices**, denoted
+$K_n$, is a simple graph with $n$ vertices and exactly one edge connecting each
+pair of distinct vertices.
+
+---
+
+Page 264
+
+**Definition**
+
+Let $m$ and $n$ be positive integers. A **complete bipartite graph on $(m, n)$
+vertices**, denoted $K_{m, n}$, is a simple graph whose vertices are divided
+into two distinct subsets, $V$ with $m$ vertices and $W$ with $n$ vertices, in
+such a way that
+
+1. every vertex of $K_{m, n}$ belongs to one of $V$ or $W$, but no vertex
+   belongs to both $V$ and $W$;
+
+2. there is exactly one edge from each vertex of $V$ to each vertex of $W$;
+
+3. there is no edge from any one vertex of $V$ to any other vertex of $V$;
+
+4. there is no edge from any one vertex of $W$ to any other vertex of $W$.

chapter_4/test_yourself.md

@@ -249,3 +249,26 @@ that integer is even; have a common factor greater than 1.
    greater than ______.
 
 $N = (2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot \dots \cdot p) + 1$; $p$
+
+---
+
+**Test Yourself**
+
+Page 265
+
+1. The total degree of a graph is defined as ______.
+
+2. The handshake theorem says that the total degree of a graph is ______.
+
+3. In any graph the number of vertices of odd degree is ______.
+
+4. A simple graph is ______.
+
+5. A complete graph on $n$ vertices is a ______.
+
+6. A complete bipartite graph on $(m, n)$ vertices is a simple graph whose
+   vertices can be divided into two distinct, non-overlapping sets, say $V$ with
+   $m$ vertices and $W$ with $m$ vertices, in such a way that (1) there is
+   ______ from each vertex of $V$ to each vertex of $W$, (2) there is ______
+   from any one vertex of $V$ to any other of $V$, and (3) there is ______ from
+   any one vertex of $W$ to any other vertex of $W$.