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Chapter 11: Problem 8

a) How many paths of length 4 are there in the complete graph \(K_{7}\) ? (Remember that a path such as \(v_{1} \rightarrow v_{2} \rightarrow\) \(v_{3} \rightarrow v_{4} \rightarrow v_{5}\) is considered to be the same as the path \(\left.v_{5} \rightarrow v_{4} \rightarrow v_{3} \rightarrow v_{2} \rightarrow v_{1} \cdot\right)\) b) Let \(m, n \in \mathbf{Z}^{+}\)with \(m

Short Answer

Expert verified
a) There are 360 paths of length 4 in a complete graph K_7. b) The total number of paths of length m in K_n is given by \( \left\lfloor \frac{n}{2} \right\rfloor \times (n-1) \times (n-2) \times... \times (n-m)\)

Step by step solution

01

Determine paths for K_7

There are 7 ways to pick the first vertex of the path in the graph. But since paths are considered the same both forward and backward, we need to take into account that each path can be counted twice. Therefore, we have \( \frac{7}{2} = 3.5 \), but since you can't have a half, it is rounded down to 3. Then for the remaining vertices, there are 6 choices for the second vertex as it cannot be the same as the first vertex. Similarly, there are 5 choices for the third vertex, and 4 choices for the fourth vertex. So, the number of paths of length 4 in K_7 is \( 3 \times 6 \times 5 \times 4 \).
02

Compute number of paths for K_7

Multiplying \( 3 \times 6 \times 5 \times 4 \), gives a total of 360 paths of length 4 in a complete graph K_7.
03

Determine paths for K_n

Using the principles from Step 1, we can find the number of paths of length m in the K_n graph. We select the integer part of \( \frac{n}{2} \) points as starting nodes. For instance, the next vertices, we have \( n-1, n-2, ..., n-m \) choices sequentially, because we can't traverse a vertex that has already been visited within a path.
04

General formula for K_n

Following the logic from previous steps, the total number of paths of length m in K_n is \( \left\lfloor \frac{n}{2} \right\rfloor \times (n-1) \times (n-2) \times... \times (n-m) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complete Graph
A complete graph is a special type of graph in discrete mathematics where every pair of distinct vertices is connected by a unique edge. It is denoted as \( K_n \), where \( n \) represents the number of vertices. In a complete graph:
  • Each vertex connects directly to every other vertex.
  • The total number of edges can be calculated using the formula \( \frac{n(n-1)}{2} \).
  • Such graphs are very dense and have the maximum possible number of edges.
In the context of the problem, we are dealing with a complete graph \( K_7 \), meaning there are 7 vertices, and each is interconnected with every other vertex. This feature of complete graphs permits a wide exploration path, which is crucial for path counting.
Path Counting
Path counting within a graph involves determining the number of distinct paths that can be formed between vertices. In a complete graph like \( K_n \), path counting becomes especially interesting due to the dense interconnections.
  • Paths are sequences of distinct vertices connected by edges.
  • For a path of length \( m \) in \( K_n \), you select \( m+1 \) distinct vertices.
  • Each path can potentially be traversed in reverse, which affects counting.
For instance, in the exercise, the challenge is to count paths of length 4 in \( K_7 \). The calculation involves selecting an initial vertex, multiplying by the choices for each subsequent vertex, and finally considering the symmetry to avoid double-counting mirror paths like \( v_1 \to v_2 \to v_3 \to v_4 \) being the same as \( v_4 \to v_3 \to v_2 \to v_1 \).
Discrete Mathematics
Discrete mathematics is a branch of mathematics dealing with discrete elements that employs algebra and arithmetic, heavily involved in graph theory.
  • It focuses on countable, distinct structures like integers, graphs, and statements in logic.
  • Graph theory, an important component, is used extensively in computer science, biology, logistics, and social sciences.
In this exercise, the principles of discrete mathematics are evident as we explore graph theory with complete graphs and the combinatorics involved in path counting. Discrete mathematics helps in constructing and analyzing algorithms, optimizing networks, and encoding information. Its role in solving problems like path counting within complete graphs is fundamental, offering insights into traversing through complex networks efficiently.

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Most popular questions from this chapter

What is the length of a longest path in each of the following graphs? a) \(K_{1,4}\) b) \(K_{3,7}\) c) \(K_{7,12}\) d) \(K_{m, n}\), where \(m, n \in \mathbf{Z}^{+}\)with \(m

Seven towns \(a, b, c, d, e, f\), and \(g\) are connected by a system of highways as follows: (1) I-22 goes from \(a\) to \(c\), passing through \(b ;\) (2) I-33 goes from \(c\) to \(d\) and then passes through \(b\) as it continues to \(f ;(3)\) I-44 goes from \(d\) through \(e\) to \(a ;\) (4) \(\mathrm{I}-55\) goes from \(f\) to \(b\), passing through \(g\); and (5) I-66 goes from \(g\). to \(d\). a) Using vertices for towns and directed edges for segments of highways between towns, draw a directed graph that models this situation. b) List the paths from \(g\) to \(a\). c) What is the smallest number of highway segments that would have to be closed down in order for travel from \(b\) to \(d\) to be disrupted? d) Is it possible to leave town \(c\) and return there, visiting each of the other towns only once? e) What is the answer to part (d) if we are not required to return to \(c ?\) f) Is it possible to start at some town and drive over each of these highways exactly once? (You are allowed to visit a town more than once, and you need not return to the town from which you started.)

a) How many paths of length 5 are there in the complete bipartite graph \(K_{3,7}\) ? (Remember that a path such as \(v_{1} \rightarrow v_{2} \rightarrow v_{3} \rightarrow v_{4} \rightarrow v_{5} \rightarrow v_{6}\) is considered to be the same as the path \(\left.v_{6} \rightarrow v_{5} \rightarrow v_{4} \rightarrow v_{3} \rightarrow v_{2} \rightarrow v_{1} .\right)\) b) How many paths of length 4 are there in \(K_{3,7}\) ? c) Let \(m, n, p \in \mathbf{Z}^{+}\)with \(2 m

Consider the complete graph \(K_{n}\) for \(n \geq 3\). Color \(r\) of the vertices in \(K_{n}\) red and the remaining \(n-r(=g)\) vertices green. For any two vertices \(v, w\) in \(K_{n}\) color the edge \(\\{v, w\\}\) (1) red if \(v, w\) are both red; (2) green if \(v, w\) are both green; or (3) blue if \(v, w\) have different colors. Assume that \(r \geq g\). a) Show that for \(r=6\) and \(g=3\) (and \(n=9\) ) the total number of red and green edges in \(K_{9}\) equals the number of blue edges in \(K_{9}\). b) Show that the total number of red and green edges in \(K_{n}\) equals the number of blue edges in \(K_{n}\) if and only if \(n=r+g\), where \(g, r\) are consecutive triangular numbers. [The triangular numbers are defined recursively by \(t_{1}=\) \(1, t_{n+1}=t_{n}+(n+1), n \geq 1 ;\) so \(t_{n}=n(n+1) / 2\). Hence \(\left.t_{1}=1, t_{2}=3, t_{3}=6, \ldots\right]\)

Let \(V=\\{a, b, c, d, e, f\\}\). Draw three nonisomorphic loop-free undirected graphs \(G_{1}=\left(V, E_{1}\right), G_{2}=\left(V, E_{2}\right)\), and \(G_{3}=\left(V, E_{3}\right)\), where, in all three graphs, we have \(\operatorname{deg}(a)=3\), \(\operatorname{deg}(b)=\operatorname{deg}(c)=2\), and \(\operatorname{deg}(d)=\operatorname{deg}(e)=\operatorname{deg}(f)=1\).
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