User:Chris/CohenProblems/Chapter 1

=Exercise 1.7.1=

The hexagon, viewed as a graph with six vertices and six edges, is bipartite.

(a) Show that the corresponding bipartition is unique.

(b) The hexagon is also tripartite. Verify that this tripartition is not unique.

Solution
(a) Without loss of generality (by symmetry), the graph $\Gamma$ of the hexagon may be viewed as the set $X = \{ 1, \ldots, 6 \}$ together with the family of subsets: $$\ast = \{ (1,2), (2,3), (3,4), (4,5), (5,6) \}.$$ The partition $X_0 = \{1,3,5 \}$, $X_1 = \{2,4,6 \}$ shows that $\Gamma$ is bipartite. To see that this decomposition is unique, let the subsets $X_0'$ and $X_1'$ of $X$ be a bipartition of $\Gamma$. The $X_i$ are non-trivial so let us assume wlog that $1 \in X_0$. Then, since $X_0$ contains no edges and because $(1,2), (1,6) \in \ast$, then we must have $2,6 \in X_1'$. Similarly $2,5 \in X_1'$ implies $3,5 \in X_0'$, which in turn gives $4,2 \in X_1'$. Thus $X_0 = \{1,3,5 \}$, $X_1 = 2,4,6$ is a unique bipartition.

(b) Under the same notation as (a), $X_0 = \{1,4 \}$, $X_2 = \{2,5 \}$, $X_2 = \{3,6 \}$ and $X_0' = \{1,5 \}$ $X_1' = \{ 2,4 \}$, $X_2 = \{3,6 \}$ are two distinct tripartitions of $X$.