A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations
We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If $L$ is a multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2 \rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable. This is a specialization of the well-known BHR Conjecture an...
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| creator | Agirseven, Onur Ollis, M. A |
| description | We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If $L$ is a
multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2
\rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable.
This is a specialization of the well-known BHR Conjecture and it includes
Buratti's original conjecture.
We argue that the most effective route to a resolution of the conjecture when
the support has size 3 is to focus on $L = \{1^a, x^b, y^c\}$, where $1 (2x^2 + 2x + 1)/(x-2)$ then the Coprime BHR
Conjecture holds for $\{1^a,x^b,y^c\}$ for infinitely many values of $v$, and
that there are at most 3 values of $v$ for which it does not hold when $(x,y) =
(6,18)$. |
| doi_str_mv | 10.48550/arxiv.2412.05750 |
| format | Article |
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multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2
\rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable.
This is a specialization of the well-known BHR Conjecture and it includes
Buratti's original conjecture.
We argue that the most effective route to a resolution of the conjecture when
the support has size 3 is to focus on $L = \{1^a, x^b, y^c\}$, where $1<x<y$,
with $a$ large subject to $a < x+y$. We use grid-based graphs to construct
linear realizations for many such multisets. A partial list of parameter sets
that the constructions cover: $a = x+y-1$; $a = x+y-2$ when $x=3$ or $x$ is
even; $a \geq 4x-3$ for $x$ odd, $y > 2x-2$, and $b \geq y-2x+2$; $a \geq x$
for $y=tx$, with $x$ and $t$ odd, and $b \geq tx+2t-3$; $a \geq 7$ for $x=3$
and $b \geq y-4$. As well as these (and further) immediate results, the
techniques introduced show promise for further development, both to head
towards a proof of the conjecture when the support has size 3 and for
situations with larger support.
We also show that if $y > (2x^2 + 2x + 1)/(x-2)$ then the Coprime BHR
Conjecture holds for $\{1^a,x^b,y^c\}$ for infinitely many values of $v$, and
that there are at most 3 values of $v$ for which it does not hold when $(x,y) =
(6,18)$.</description><identifier>DOI: 10.48550/arxiv.2412.05750</identifier><language>eng</language><subject>Mathematics - Combinatorics</subject><creationdate>2024-12</creationdate><rights>http://creativecommons.org/licenses/by-nc-sa/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2412.05750$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2412.05750$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Agirseven, Onur</creatorcontrib><creatorcontrib>Ollis, M. A</creatorcontrib><title>A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations</title><description>We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If $L$ is a
multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2
\rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable.
This is a specialization of the well-known BHR Conjecture and it includes
Buratti's original conjecture.
We argue that the most effective route to a resolution of the conjecture when
the support has size 3 is to focus on $L = \{1^a, x^b, y^c\}$, where $1<x<y$,
with $a$ large subject to $a < x+y$. We use grid-based graphs to construct
linear realizations for many such multisets. A partial list of parameter sets
that the constructions cover: $a = x+y-1$; $a = x+y-2$ when $x=3$ or $x$ is
even; $a \geq 4x-3$ for $x$ odd, $y > 2x-2$, and $b \geq y-2x+2$; $a \geq x$
for $y=tx$, with $x$ and $t$ odd, and $b \geq tx+2t-3$; $a \geq 7$ for $x=3$
and $b \geq y-4$. As well as these (and further) immediate results, the
techniques introduced show promise for further development, both to head
towards a proof of the conjecture when the support has size 3 and for
situations with larger support.
We also show that if $y > (2x^2 + 2x + 1)/(x-2)$ then the Coprime BHR
Conjecture holds for $\{1^a,x^b,y^c\}$ for infinitely many values of $v$, and
that there are at most 3 values of $v$ for which it does not hold when $(x,y) =
(6,18)$.</description><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMjE00jMwNTc14GTwclRwzi8oysxNVXAqLUosKcnU9cgvSszWDcovTgRK5WWlJpeUFqUqJOalKLgXZaboOiUWp6Yo-GTmpSYWKQSlJuZkViWWZObnFfMwsKYl5hSn8kJpbgZ5N9cQZw9dsK3xIEsSiyrjQbbHg203JqwCAAMcOaI</recordid><startdate>20241207</startdate><enddate>20241207</enddate><creator>Agirseven, Onur</creator><creator>Ollis, M. A</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20241207</creationdate><title>A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations</title><author>Agirseven, Onur ; Ollis, M. A</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2412_057503</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Agirseven, Onur</creatorcontrib><creatorcontrib>Ollis, M. A</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Agirseven, Onur</au><au>Ollis, M. A</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations</atitle><date>2024-12-07</date><risdate>2024</risdate><abstract>We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If $L$ is a
multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2
\rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable.
This is a specialization of the well-known BHR Conjecture and it includes
Buratti's original conjecture.
We argue that the most effective route to a resolution of the conjecture when
the support has size 3 is to focus on $L = \{1^a, x^b, y^c\}$, where $1<x<y$,
with $a$ large subject to $a < x+y$. We use grid-based graphs to construct
linear realizations for many such multisets. A partial list of parameter sets
that the constructions cover: $a = x+y-1$; $a = x+y-2$ when $x=3$ or $x$ is
even; $a \geq 4x-3$ for $x$ odd, $y > 2x-2$, and $b \geq y-2x+2$; $a \geq x$
for $y=tx$, with $x$ and $t$ odd, and $b \geq tx+2t-3$; $a \geq 7$ for $x=3$
and $b \geq y-4$. As well as these (and further) immediate results, the
techniques introduced show promise for further development, both to head
towards a proof of the conjecture when the support has size 3 and for
situations with larger support.
We also show that if $y > (2x^2 + 2x + 1)/(x-2)$ then the Coprime BHR
Conjecture holds for $\{1^a,x^b,y^c\}$ for infinitely many values of $v$, and
that there are at most 3 values of $v$ for which it does not hold when $(x,y) =
(6,18)$.</abstract><doi>10.48550/arxiv.2412.05750</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations |
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