arXiv subjects

Jonathan Chappelon

Publications and source records attributed to Jonathan Chappelon.

22 records · Page 2Linked to original sources

arXiv2014

Ramsey for complete graphs with dropped cliques

Let $K\_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K\_{[k\_1,t\_1]},\dots, K\_{[k\_r,t\_r]})$ (the smallest integer $n$ such that for any $r$-edge coloring of $K\_n$ there always occurs a monochromatic $K\_{[k\_i,t\_i]}$ for some $i$). Our first upper bound contains a classical one in the case when $k\_1=\cdots =k\_r$ and $t\_i=1$ for all $i$. The second one is obtained by introducing a new edge coloring called {\em $χ\_r$-colorings}. We finally discuss a conjecture claiming, in particular, that our second upper bound improves the classical one in infinitely many cases.

math.CO↗

arXiv2013

Modular Schur numbers

For any positive integers l and m, a set of integers is said to be (weakly) l-sum-free modulo m if it contains no (pairwise distinct) elements $x_1,x_2,...,x_l,y$ satisfying the congruence $x_1+\...+x_l\equiv y\bmod{m}$. It is proved that, for any positive integers k and l, there exists a largest integer $n$ for which the set of the first $n$ positive integers $\{1,2,\...,n\}$ admits a partition into k (weakly) l-sum-free sets modulo m. This number is called the generalized (weak) Schur number modulo $m$, associated with k and l. In this paper, for all positive integers k and l, the exact value of these modular Schur numbers are determined for m=1, 2 and 3.

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arXiv2009

Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size $n$ which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy $a_{i,j}=a_{i-1,j-1}+a_{i-1,j}$ for all $2\leq i<j\leq n$. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices $K_2$ is the only regular Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if $(a_{i,j})_{1\leq i,j\leq n}$ is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix $(a_{i,j})_{2\leq i,j\leq n-1}$ is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size $n$ whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on $\lceil \frac{n}{24}\rceil$ parameters for all even numbers $n$, and on $\lceil \frac{n}{30}\rceil$ parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to 1500 vertices in the odd case.

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arXiv2008

On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$. The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length 1 is reached, determines a triangle, the \textit{Steinhaus triangle $ΔX$ generated by the sequence $X$}. We say that $X$ is \textit{balanced} if its Steinhaus triangle $ΔX$ contains each element of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. An obvious necessary condition for $m$ to be the length of a balanced sequence in $\mathbb{Z}/n\mathbb{Z}$ is that $n$ divides the binomial coefficient $\binom{m+1}{2}$. It is an open problem to determine whether this condition on $m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for $n=2$ and generalized by John C. Molluzzo in 1976 for $n\geq3$. So far, only the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we answer positively Molluzzo's problem in the case $n=3^k$ for all $k\geq1$. Moreover, for every odd integer $n\geq3$, we construct infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. This is achieved by analysing the Steinhaus triangles generated by arithmetic progressions. In contrast, for any $n$ even with $n\geq4$, it is not known whether there exist infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. As for arithmetic progressions, still for $n$ even, we show that they are never balanced, except for exactly 8 cases occurring at $n=2$ and $n=6$.

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