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Register a new userGrowable Realizations: a Powerful Approach to the Buratti-Horak-Rosa Conjecture
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Label the vertices of the complete graph $K_v$ with the integers ${ 0, 1, ldots, v-1 }$ and define the length of the edge between $x$ and $y$ to be $min( |x-y| , v - |x-y| )$. Let $L$ be a multiset of size $v-1$ with underlying set contained in ${ 1, ldots, lfloor v/2 rfloor }$. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in $K_v$ whose edge lengths are exactly $L$ if and only if for any divisor $d$ of $v$ the number of multiples of $d$ appearing in $L$ is at most $v-d$. We introduce growable realizations, which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in ${ 1,4,5 }$ or in ${ 1,2,3,4 }$ and a partial result when the underlying set has the form ${ 1, x, 2x }$. We believe that for any set $U$ of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set $U$.
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The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list $L$ of $v-1$ positive integers not exceeding $leftlfloor frac{v}{2}rightrfloor$ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set ${0,1,ldots,v-1}$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $v-d$. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: ${x,y,x+y}$, ${1,2,3,4}$, ${1,2,4,ldots,2x}$, ${1,2,4,ldots,2x,2x+1}$. We also consider lists with many consecutive elements.
An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,mathbb{F}_p)$ over the field $mathbb{F}_p := mathbb{Z}/pmathbb{Z}$ is the unique projective plane of order $p$. Let $pi$ be any projective plane of order $p$. For any partial linear space ${cal X}$, define the inclusion number $i({cal X},pi)$ to be the number of isomorphic copies of ${cal X}$ in $pi$. In this paper we prove that if ${cal X}$ has at most $log_2 p$ lines, then $i({cal X},pi)$ can be written as an explicit rational linear combination (depending only on ${cal X}$ and $p$) of the coefficients of the complete weight enumerator (c.w.e.) of the $p$-ary code of $pi$. Thus, the c.w.e. of this code carries an enormous amount of structural information about $pi$. In consequence, it is shown that if $p > 2^ 9=512$, and $pi$ has the same c.w.e. as $PG(2,mathbb{F}_p)$, then $pi$ must be isomorphic to $PG(2,mathbb{F}_p)$. Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture by using induction with respect to the degree of the polynomials.
This report formulates a conjectural combinatorial rule that positively expands Grothendieck polynomials into Lascoux polynomials. It generalizes one such formula expanding Schubert polynomials into key polynomials, and refines another one expanding stable Grothendieck polynomials.
In 1990, Cvetkovi{c} and Rowlinson [The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that among all outerplanar graphs on $n$ vertices, $K_1vee P_{n-1}$ attains the maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed the conjecture for sufficiently large values of $n$. In this article, we show the conjecture is true for all $ngeq2$ except for $n=6$.
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