Grid-Based Graphs, Linear Realizations and the Buratti-Horak-Rosa Conjecture

Abstract

Label the vertices of the complete graph $K_v$ with the integers $\{0, 1, \ldots, v-1\}$ and define the {\em length} $\ell$ of the edge between distinct vertices labeled $x$ and $y$ by $\ell(x,y) = \min( |y-x|, v - |y-x| )$. A {\em realization} of a multiset $L$ of size $v-1$ is a Hamiltonian path through $K_v$ whose edge labels are $L$. The {\em Buratti-Horak-Rosa (BHR) Conjecture} is that there is a realization for a multiset $L$ if and only if for any divisor $d$ of $v$ the number of multiples of $d$ in $L$ is at most $v-d$. We introduce ``grid-based graphs" as a useful tool for constructing particular types of realizations, called ``linear realizations," especially when the multiset in question has a support of size 3. This lets us prove many new instances of the BHR Conjecture, including those for multisets of the form $\{1^a, x^b, y^c \}$ when $a \geq x+y - ε$, where $ε$ is the number of even elements in $\{ x,y \}$, and those for all multisets of the following forms for sufficiently large $v$ with $\gcd(v,y) = 1$ for all $y \in L$: $\{1^a, 2^b, x^c\}$, except possibly when $a \in \{1,2\}$ and $x$ is odd, $\{1^a, x^b, (x+1)^c\}$. This establishes that there are infinitely many sets $U$ of size 3 for which there are infinitely many values of $v$ where the BHR Conjecture holds for each multiset with support $U$. We also show that the BHR Conjecture holds for $\{1^a,x^b,(x+1)^c\}$ when $x \in \{7,9,10\}$ and $\gcd(v,x) = \gcd(v,x+1) = 1$.

Figures (14)

• Figure 1: Linear realizations for $\{1^6,7^{18}\}$ and $\{ 1^7, 7^{17}, 8^3\}$.
• Figure 2: Linear realizations of $\{1^{3}, 2^5, 9^{24}\}$ and $\{1^2, 2^5, 8^{21} \}$, constructed via Lemmas \ref{['lem:2rect']} and \ref{['lem:2squash']} respectively.
• Figure 3: The perfect linear realizations $\mathbf{g_1}$ and $\mathbf{g_2}$ for $\{ (x-1)^{x-1} , x^{x} \}$ and $\{ x^{x+2} , (x+1)^{x-1} \}$ respectively.
• Figure 4: The standard linear realizations $\mathbf{h_1}$ (first diagram) and $\mathbf{h_2}$ (remaining diagrams) of $\{1^{x-1}, x^b\}$ for $(v,x,b) = (25, 7,18)$, $(29, 8,21)$, $(12, 7, 5)$ and $(22, 7,15)$.
• Figure 5: The standard linear realizations $\mathbf{h_3}$ and $\mathbf{h_4}$ of $\{1^{x}, x^b\}$ for $(v,x,b) = (25, 7,17)$.

Theorems & Definitions (44)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof