European Journal of Combinatorics最新文献

We show that the chronological removal of cycles from a walk on a graph, known as Lawler’s loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.

We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from $\{1,\dots n\}$ be such that, for every pair of subsets in the family, the intersection contains a sum $x+y=z$? We conjecture that any such sum-intersecting family must have size at most $\frac{1}{4}\cdot 2^n$ (which would be tight if correct). Towards this conjecture, we show that every sum-intersecting family has at most $0.32\cdot 2^n$ subsets.

The families $F_1\subseteq \left(\frac{\left[n\right]}{k_1}\right),F_2\subseteq \left(\frac{\left[n\right]}{k_2}\right),\dots ,F_r\subseteq \left(\frac{\left[n\right]}{k_r}\right)$ are said to be cross-intersecting if $|F_i\cap F_j|⩾1$ for any $1⩽i<j⩽r$ and $F_i\in F_i$, $F_j\in F_j$. Cross-intersecting families $F_1,F_2,\dots ,F_r$ are said to be non-empty if $F_i\ne 0̸$ for any $1⩽i⩽r$. This paper shows that if $F_1\subseteq \left(\frac{\left[n\right]}{k_1}\right),F_2\subseteq \left(\frac{\left[n\right]}{k_2}\right),\dots ,F_r\subseteq \left(\frac{\left[n\right]}{k_r}\right)$ are non-empty cross-intersecting families with