Exam 10: A: Graphs

find $\kappa ( G ) , \lambda ( G ) \text {, and } \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text {, and determine which of the two inequalities }$ $\text { in } \kappa ( G ) \leq \lambda ( G ) \leq \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text { are strict }$ -

fill in the blanks. -The vertex-chromatic number for $K _ { 7,7 }$ = ____.

fill in the blanks. -List all positive integers n such that $K _ { n }$ has a Hamilton circuit ____.

If a simple graph G has $\ v$ vertices and e edges, how many edges does $\overline G$ have?

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find the vertex-chromatic number for $G _ { m, n }$ .

Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path?

fill in the blanks. -List all the positive integers n such that $K _ { 2 , n }$ is planar.____

fill in the blanks. -The length of the longest simple circuit in K_4,10 is ____.

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -For which positive integers m and n does $G _ { m , n }$ have a Hamilton circuit?

fill in the blanks. -The vertex-chromatic number for $W _ { 9 }$ = ____.

find $\kappa ( G ) , \lambda ( G ) \text {, and } \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text {, and determine which of the two inequalities }$ $\text { in } \kappa ( G ) \leq \lambda ( G ) \leq \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text { are strict }$ -

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find the edge-chromatic number for $G _ { m, n }$ .

In K₅ find the number of paths of length 2 between every pair of vertices.

Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path?

fill in the blanks. -Every Hamilton circuit for $W _ { n }$ has length ____.

Solve the traveling salesman problem for the graph at the right by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight.

either give an example or prove that there are none. -A simple digraph with indegrees 0, 1, 2, 2 and outdegrees 0, 1, 1, 3.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K₄.

fill in the blanks. -The incidence matrix for Q5 has rows ____ and ____ columns.

In the group stage of the 2011 women's soccer world cup the USA beat North Korea, Sweden beat Columbia, the USA beat Columbia, Sweden beat North Korea, Sweden beat the USA, and the game between Columbia and North Korea ended in a tie. Model this outcome using a directed segment from A to B if A beat B, and an undirected segment if the game ended in a tie.