Combinatorics (math.CO)

Abstract: Let $G$ be a planar graph and let $C$ be a cycle in $G$. Inside of each finite face of $G$, we write down the number of edges of that face which belong to $C$. This is the signature of $C$ in $G$. The notion of a signature arises naturally in the context of Slitherlink puzzles. The signature of a cycle does not always determine it uniquely. We focus on the ambiguity of signatures in the case when $G$ is a rectangular grid of unit square cells. We describe all grids which admit an ambiguous signature. For each such grid, we then determine the greatest possible difference between two cycles with the same signature on it. We also study the possible values of the total number of cycles which fit a given signature. We discuss various related questions as well.

Abstract: We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ and for every infinite $\mathcal{M} \subset \mathbb{R}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $\mathcal{M} \subset \mathbb{R}^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $\mathcal{M}$.

Abstract: We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha+o(1)}$.

Abstract: We study two notions of fan-planarity introduced by (Cheong et al., GD22), called weak and strong fan-planarity that separate two non-equivalent definitions of fan-planarity in the literature. We prove that not every weakly fan-planar graph is strongly fan-planar, while the density upper bound for both families is the same.

Abstract: In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs.

Abstract: The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$.

Abstract: In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group $\C$ and a subgroup $N \leq \C$. Subgroups $N \leq \C$ that give rise to abstract polytopes through such construction are called {\em sparse}. If, further, the stabilizer of a base flag of the poset is precisely $N$, then $N$ is said to be {\em semisparse}. In \cite[Conjecture 5.2]{hartley1999more} Hartley conjectures that sparse groups are always semisparse. In this paper, we show that this conjecture is in fact false: there exist sparse groups that are not semisparse. In particular, we show that such groups are always obtained from non-faithful maniplexes that give rise to polytopes. Using this, we show that Hartely's conjecture holds for rank 3, but we construct examples to disprove the conjecture for all ranks $n\geq 4$.

Abstract: In this article the cyclic cutwidth of the $n$-dimensional cube is explored. It has been conjectured by Dr. Chavez and Dr. Trapp that the cyclic cutwidth of $Q_n$ is minimized with the Graycode numbering. Several results have been found toward the proof of this conjecture.

Abstract: Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.