Question 39
We are given a connected graph with $n$ edges, in which any two vertices are joined by at most one edge. For any two cycles $C$ and $C'$ of this graph, define their **outer cycle** as
$$
C \star C' = \{x \mid x \in (C - C') \cup (C' - C)\}.
$$
(1) Let $r$ be the largest positive integer such that we can choose $r$ cycles $C_1, C_2, \dots, C_r$ from this graph such that for all $1 \le k \le r$ and all $1 \le i, j_1, j_2, \dots, j_k \le r$ with $i \notin \{j_1, \dots, j_k\}$, we have
$$
C_i \neq C_{j_1} \star C_{j_2} \star \dots \star C_{j_k}.
$$
(2) Let $s$ be the largest positive integer such that we can choose $s$ of these $n$ edges so that the chosen edges contain no cycle.
Prove that $r+s=n$.
Remark: a cycle is a set of edges of the form $\{A_i A_{i+1}\}, 1 \le i \le n$, where $n \ge 3$, $A_1, A_2, \dots, A_n$ are distinct vertices, and $A_{n+1} = A_1$.
$$
C \star C' = \{x \mid x \in (C - C') \cup (C' - C)\}.
$$
(1) Let $r$ be the largest positive integer such that we can choose $r$ cycles $C_1, C_2, \dots, C_r$ from this graph such that for all $1 \le k \le r$ and all $1 \le i, j_1, j_2, \dots, j_k \le r$ with $i \notin \{j_1, \dots, j_k\}$, we have
$$
C_i \neq C_{j_1} \star C_{j_2} \star \dots \star C_{j_k}.
$$
(2) Let $s$ be the largest positive integer such that we can choose $s$ of these $n$ edges so that the chosen edges contain no cycle.
Prove that $r+s=n$.
Remark: a cycle is a set of edges of the form $\{A_i A_{i+1}\}, 1 \le i \le n$, where $n \ge 3$, $A_1, A_2, \dots, A_n$ are distinct vertices, and $A_{n+1} = A_1$.