### $(-1)$-enumeration of self-complementary plane partitions.

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An $n\times n$ sign pattern $\mathcal{A}$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $\mathcal{A}$. Let ${\mathcal{D}}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$$(i=1...$

As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the removal of the internal vertices and edges of the path that is the intersection of...

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The simple incidence structure $\mathcal{D}(\mathcal{A},2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane $\mathcal{A}=(\mathcal{P},\mathcal{L})$ of order $n>2$ is a $2-({n}^{2},2n,2n-1)$ design. If $n=3$, $\mathcal{D}(\mathcal{A},2)$ is the complementary design of $\mathcal{A}$. If $n=4$, $\mathcal{D}(\mathcal{A},2)$ is isomorphic to the geometric design $A{G}_{3}(4,2)$ (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a $2-({n}^{2},2n,2n-1)$ design to be of the form $\mathcal{D}(\mathcal{A},2)$ for some finite affine plane $\mathcal{A}$ of order $n>4$. As a consequence we obtain a characterization of small designs $\mathcal{D}(\mathcal{A},2)$.

The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.