(a) Recall that the path graph on n vertices has vertex set V = {1,.., n} and edge set E = {{i,i+1} : i = 1, .,n – 1}. How many 3-colorings does the path graph on n vertices have? (hint: use a recurrence relation). (b) Let k be a positive integer. Show that every tree on n vertices has exactly the same number of k-colorings. What is this number of colorings? (hint: consider your answers to previous parts of this question)

(a) Recall that the path graph on n vertices has vertex set V = {1,.., n} and edge set E = {{i,i+1} : i = 1, .,n – 1}. How many 3-colorings does the path graph on n vertices have? (hint: use a recurrence relation). (b) Let k be a positive integer. Show that every tree on n vertices has exactly the same number of k-colorings. What is this number of colorings? (hint: consider your answers to previous parts of this question)

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

(a) Recall that the path graph on n vertices has vertex set V ={1,.., n} and edge set
E = {{i,i+1} : i = 1, . . . , n – 1}. How many 3-colorings does the path graph on n
vertices have? (hint: use a recurrence relation).
(b) Let k be a positive integer. Show that every tree on vertices has exactly the same number
of k-colorings. What is this number of colorings? (hint: consider your answers to previous parts
of this question)

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