Construction Techniques for Linear Realizations of Multisets with Small Support

TL;DR

The paper advances the construction of linear realizations for multisets of edge-lengths, focusing on supports of size 3 of the form $\{1^a, (y-k)^b, y^c\}$. It develops two complementary construction streams: core perfect linear realizations for coprime pairs $(x,y)$ and spanning-forest based standard realizations for small $k$, coupled with $\omega$-constructions and growability concepts to connect pieces. For $k=1$, it completes the $y\le 16$ case of the Coprime BHR Conjecture and provides new small-$b$/$c$ methods extending the range where realizations exist; for $k=2$, it proves standard realizations exist when $a\ge y$ (with parity-sensitive refinements) and introduces beta-moves and trapezoid traversals to manage edge-type switches. Overall, the work refines a general framework for constructing linear realizations that support progress toward the Coprime BHR Conjecture for supports of size 3 and points toward extensions to arbitrary $k$.

Abstract

A Hamiltonian path in the complete graph $K_v$ whose vertices are labeled with the integers $0,1,\ldots,v-1$ is a linear realization for the multiset $L$ of the linear edge-lengths (given by $|x-y|$ for the edge between vertices $x$ and $y$) of the edges in the path. A linear realization is standard if an end-vertex is 0 and perfect if the end-vertices are 0 and $v-1$. Linear realizations are useful in the study of the Buratti-Horak-Rosa (BHR) Conjecture on the existence of cyclic realizations (where cyclic edge-lengths are given by distance modulo $v$) for given multisets. In this paper, we focus on multisets of the form $\{1^a, (y-k)^b, y^c\}$. Using core perfect linear realizations for supports of size 2 (which have the forms $\{x^{y-1},y^{x+1}\}$ whenever $\gcd(x,y)=1$), we construct standard linear realizations (with $a=k-1$, $b=j(y-k)$, $c=jy$) when $k\mid y$ or $k \leq 4$. When $k=2$, these allow us to show that there is a linear realization whenever $a \geq y$. This is in line with the known results for the case of $k=1$. We also supplement these results for $k=1$ by constructing linear realizations whenever $b+c < y$ and $a \geq y - \min(b,c)$, from which the coprime version of the BHR Conjecture (requiring that $v$ is coprime with each element of the multiset) follows for $k=1$ when $y \leq 16$. Our methods show promise for constructing linear realizations for arbitrary $k$, in the direction of a resolution of the BHR Conjecture for supports of size 3.

Figures (21)

• Figure 1: Standard linear realizations $\mathbf{h_1}$ and $\mathbf{h_2}$ for $\{1^{x-1}, x^b\}$ for $(v,x,b) = (25, 7,18)$, $(29, 8,21)$, $(12, 7, 5)$ and $(22, 7,15)$AO1.
• Figure 2: Standard linear realizations of $\{1^{x}, x^b\}$ for $(v,x,b) = (25, 7,17)$, $(29, 8,20)$, $(12, 7, 4)$ and $(22, 7,14)$ obtained from those of Figure \ref{['fig:omega1']} by applying a tail curl.
• Figure 3: Examples of type $\mathcal{C}_y$ standard linear realizations for $\{1^6, 7^{7}\}$ and $\{1^8, 8^2\}$.
• Figure 4: Examples of 7-growable standard linear realizations for $\{1^7, 7^{16}\}$ and $\{1^7, 7^{14}\}$.
• Figure 5: A perfect linear realization for $\{ 5^7, 6^4\}$ and a standard linear realization for $\{ 1 , 4^4 , 6^6 \}$.

Theorems & Definitions (52)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Example 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Example 1.10