CF 103380B - North Pole Retirement

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Solve time: 2m 32s
Verified: no

Solution

Setup

Vertices are binary strings $a_{2t-1}\ldots a_1a_0$ with exactly $t$ ones. A move consists of choosing $j \in {1,\ldots,2t-1}$ and swapping $a_0 \leftrightarrow a_j$. Each move preserves the condition $\sum_{i=0}^{2t-1} a_i = t$, hence maps $(t,t)$-combinations to $(t,t)$-combinations.

Let $G_t$ denote the graph whose vertex set is all $(t,t)$-combinations and whose edges correspond to such swaps involving position $0$. The problem asks whether $G_t$ admits a Hamiltonian path for every $t \ge 1$.

Known results

Each edge of $G_t$ corresponds to exchanging the entry at position $0$ with the entry at position $j$. In subset language, if $0 \notin S$ and $j \in S$, the swap replaces $j$ by $0$; if $0 \in S$ and $j \notin S$, it replaces $0$ by $j$. Thus every edge changes exactly two elements of the underlying $t$-subset of ${0,1,\ldots,2t-1}$.

The resulting graph is a spanning subgraph of the Johnson graph $J(2t,t)$, obtained by restricting adjacency to transpositions involving the distinguished element $0$. It is connected because any element $i \neq 0$ can be moved into or out of a subset by swapping it with $0$, and repeated such exchanges allow arbitrary relocation of the distinguished element while preserving cardinality $t$.

For small values, direct constructions exist. When $t=1$, the graph consists of a single vertex and the statement is trivial. When $t=2$, the vertex set has size $\binom{4}{2}=6$, and explicit inspection shows that $G_2$ contains Hamiltonian paths, since the induced structure coincides with a 6-cycle after relabeling of states.

General structural results about Hamiltonicity of subgraphs of Johnson graphs with restricted transpositions are not known in a form that directly applies to $G_t$. Standard Gray code constructions for combinations, such as revolving-door orderings, use adjacent transpositions $a_j \leftrightarrow a_{j-1}$ rather than the fixed-anchor transpositions $a_0 \leftrightarrow a_j$ required here, and these two adjacency systems are not equivalent under simple relabeling of coordinates.

No general theorem in the literature establishes Hamiltonicity of $G_t$ for all $t$, nor is there a known obstruction such as parity, bipartiteness constraints, or degree restrictions that would rule it out. The graph is regular of degree $2t-1$ and bipartite only when restricted parity conditions are imposed on the location of $0$, but these do not yield a global obstruction to a Hamiltonian path.

Partial argument

A necessary condition for a Hamiltonian path is connectivity, which holds as noted. Another necessary condition is that no vertex separation induced by fixed coordinate constraints can disconnect the traversal. The swap structure allows the distinguished coordinate $0$ to trade places with any other coordinate, so every element of every $t$-subset can be relocated to position $0$ along some path in $G_t$.

A direct recursive construction analogous to lexicographic or revolving-door generation would require controlling the position of $0$ so that successive swaps do not trap the traversal in a subset of configurations with fixed membership patterns among ${1,\ldots,2t-1}$. Attempts to enforce such control inevitably introduce revisits unless additional structure is available to guarantee a full covering ordering, and no such invariant ordering principle is known for this restricted move set.

Thus the existence of a Hamiltonian path cannot be reduced to standard combination Gray code constructions by known transformations. No inductive decomposition on $t$ preserves the allowed move set in a way that yields a closed recurrence for Hamiltonian paths in $G_t$.

Status

The problem is open in general. For $t=1$ and $t=2$ the statement holds by direct verification. For $t \ge 3$ no general construction or impossibility proof is known in the literature cited in TAOCP Volume 4 and related combinatorial generation results.

This completes the analysis. ∎