From e6703ca90b8f4c2a39a1d5a05a5a694b12ad125b Mon Sep 17 00:00:00 2001 From: tomit4 Date: Sun, 14 Jun 2026 15:26:14 -0700 Subject: [PATCH] :construction: Setup for 4.9 --- chapter_4/exercises.md | 140 +++++++++++++++++++++++++++++++++++++ chapter_4/notes.md | 132 ++++++++++++++++++++++++++++++++++ chapter_4/test_yourself.md | 23 ++++++ 3 files changed, 295 insertions(+) diff --git a/chapter_4/exercises.md b/chapter_4/exercises.md index 1e7d7e0..3f0d18a 100644 --- a/chapter_4/exercises.md +++ b/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. diff --git a/chapter_4/notes.md b/chapter_4/notes.md index c7b87a7..3f2f165 100644 --- a/chapter_4/notes.md +++ b/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 $W$. diff --git a/chapter_4/test_yourself.md b/chapter_4/test_yourself.md index e7c1897..b9dac7d 100644 --- a/chapter_4/test_yourself.md +++ b/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 $W$.