Hey Ben, great work testing your algorithm across various graphs—let's dive into a counterexample that should trip up your greedy approach.

你的算法步骤回顾 #

To make sure we're on the same page, here's a clear breakdown of your approach:

• Draw the undirected graph $G$, ensuring all nodes touch the exterior region surrounding $G$
• Greedily number the internal regions formed by the graph's edges as $R=1,2,...,m-1$, and label the exterior region as $m$
• For each node $N$, generate a label $C_N$: a number formed by the ordered set of all region numbers (including the exterior $m$) adjacent to $N$
• Start at a random node
• Move to the adjacent, unvisited node with the smallest $C_N$; repeat until you can't continue
• Check both ends of the current path—if one end can still expand, restart step 5 from that end
• If all nodes are visited, conclude a Hamiltonian path exists; otherwise, conclude it doesn't

反例构造：分支陷阱图 #

We can build a graph where your greedy selection gets stuck in a small branch early on, even though a valid Hamiltonian path exists. Here's the structure:

1. A central node $X$ connected to four branch nodes: $A, B, C, D$
2. Branch $A$ has two leaf nodes: $A_1$ and $A_2$ (both connected only to $A$)
3. Branch $B$ has one leaf node: $B_1$ (connected only to $B$)
4. Branch $C$ has one leaf node: $C_1$ (connected only to $C$)
5. Branch $D$ has two leaf nodes: $D_1$ and $D_2$ (both connected only to $D$)

区域编号与$C_N$计算

• Internal regions: Each branch's "small triangular" area gets a unique number: $R_1$ (for $A-X-A_1$), $R_2$ (for $A-X-A_2$), $R_3$ (for $B-X-B_1$), $R_4$ (for $C-X-C_1$), $R_5$ (for $D-X-D_1$), $R_6$ (for $D-X-D_2$)
• Exterior region $m=7$
• Node labels ($C_N$):
  • $A_1$: 17 (adjacent to $R_1$ + exterior 7)
  • $A_2$: 27 (adjacent to $R_2$ + exterior 7)
  • $A$: 127 (adjacent to $R_1, R_2$ + exterior 7)
  • $B_1$: 37 (adjacent to $R_3$ + exterior 7)
  • $B$: 37 (adjacent to $R_3$ + exterior 7)
  • $C_1$: 47 (adjacent to $R_4$ + exterior 7)
  • $C$: 47 (adjacent to $R_4$ + exterior 7)
  • $D_1$: 57 (adjacent to $R_5$ + exterior 7)
  • $D_2$: 67 (adjacent to $R_6$ + exterior 7)
  • $D$: 567 (adjacent to $R_5, R_6$ + exterior 7)
  • $X$: 1234567 (adjacent to all internal regions + exterior 7)

算法执行过程

Let's start at node $A_1$:

1. First move: $A_1$ only has one unvisited neighbor ($A$, $C_N=127$), so path becomes $A_1 → A$
2. Second move: $A$ has two unvisited neighbors—$X$ ($C_N=1234567$) and $A_2$ ($C_N=27$). Your algorithm picks the smaller $C_N$, so path becomes $A_1 → A → A_2$
3. Stuck: $A_2$ has no unvisited neighbors. Checking both ends of the path, $A_1$ also has no unvisited neighbors left
4. The algorithm incorrectly concludes no Hamiltonian path exists, but a valid path does exist (e.g., $A_1 → A → X → B → B_1 → C → C_1 → D → D_1 → D_2$)

This graph is a solid counterexample: your greedy focus on the smallest adjacent $C_N$ traps you in the $A$ branch early, preventing you from accessing the rest of the graph even though a Hamiltonian path is present.

备注：内容来源于stack exchange，提问作者Ben