Related papers: The Buratti-Horak-Rosa Conjecture Holds for Some U…

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…

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…

The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset L of v-1 positive integers not exceeding [v/2] is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with…

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…

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 $\left\lfloor \frac{v}{2}\right\rfloor$ is the list of edge-lengths of a suitable…

This paper presents a comprehensive computational approach to verify and inductively construct Hamiltonian paths for the Buratti--Horak--Rosa (BHR) Conjecture. The conjecture posits that for any multiset $L$ of $p-1$ positive integers not…

Marco Buratti's conjecture states that if $p$ is a prime and $L$ a multiset containing $p-1$ non-zero elements from the integers modulo $p$, then there exists a Hamiltonian path in the complete graph of order $p$ with edge lengths in $L$.…

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…

In this paper we investigate a problem proposed by Marco Buratti, Peter Horak and Alex Rosa (denoted by BHR-problem) concerning Hamiltonian paths in the complete graph with prescribed edge-lengths. In particular we solve BHR({1^a,2^b,t^c})…

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with $1,2,3$ so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in…

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karo\'nski, {\L}uczak, and Thomason in 2004: for every connected graph different from $K_2$, we can assign labels~$1,2,3$ to the edges so that no two adjacent…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…

A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…

The conjecture of Brown, Erd\H{o}s and S\'os from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an $r$-uniform hypergraph ($r$-graph) $G$, let…

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of…

A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the…

In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total…