Construction Techniques for Linear Realizations of Multisets with Small Support

TL;DR

This paper develops construction techniques for linear realizations of specific multisets, advancing understanding of the Buratti-Horak-Rosa Conjecture and providing new methods for supports of size 3.

Contribution

It introduces new construction methods for linear realizations of multisets with small support, especially for cases where the multiset has a specific form and parameters.

Findings

Constructed linear realizations for multisets with support size 2 when certain divisibility conditions hold.

Established conditions under which linear realizations exist for multisets with support size 2, including the case when k=2.

Provided partial results supporting the coprime version of the BHR Conjecture for y ≤ 16.

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.