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It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $ngeq 4$. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph of $n$-vertex flag $2$-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph of $n$-vertex stacked $2$-spheres has at least as many connected components as there are trees on $lfloorfrac{n-5}{3}rfloor$ nodes with maximum node-degree at most four.
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We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Delta$, we show that its $gamma$-vector $gamma^Delta=(1,gamma_1,gamma_2,ldots)$ satisfies: begin{align*} gamma_j=0,text{ for all } j>gamma_1, quad gamma_2leqbinom{gamma_1}{2}, quad gamma_{gamma_1}in{0,1}, quad text{ and }gamma_{gamma_1-1}in{0,1,2,gamma_1}, end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Delta$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $gamma$-vectors of flag homology spheres (of any dimension); these are the first examples of this kind. In addition, we prove a flag analog of Perles 1970 theorem on $k$-skeleta of polytopes with few vertices, specifically: the number of combinatorial types of $k$-skeleta of flag homology spheres with $gamma_1leq b$, of any given dimension, is bounded independently of the dimension.
A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some graph on a pair of pants. This characterizes the image of a purely topological version of the spectral map for the vector bundle Laplacian for a flat connection on a pair of pants. The proof uses a graph exploration technique akin to the peeling of planar maps.
The sequence $(x_n)_{ninmathbb N} = (2,5,15,51,187,dots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$ with four different letters. It is also the cardinality of the quotient of $(mathbb Z_2times mathbb Z_2)^n$ under the left action of the special linear group $mathrm{SL}(2,mathbb Z)$. In this paper we show how these two interpretations of $x_n$ are related to each other. More generally, for prime numbers $p$ we show a correspondence between a quotient of $(mathbb Z_ptimesmathbb Z_p)^n$ and a language with $p^2$ letters and words of length $n$.
Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $Ssubseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $partial_{2p}(G)=max{partial(S): Ssubseteq V(G) text{ is a }2text{-packing}}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of $2$-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $partial_{2p}(G)+mu_{_R}(G)=n(G)$, where $mu_{_R}(G)$ denotes the unique response Roman domination number of $G$. As a consequence of the study, we derive several combinatorial results on $mu_{_R}(G)$, and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem.
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