The Crossing Number of Circulant Graph C(3k+1;{1,k}) on the Projective Plane

The regularity of some families of circulant graphs

We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.

Achromatic number, achromatic index and diachromatic number of circulant graphs and digraphs

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $α(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the lower bounds $α(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and $dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\m…

The crossing number of the generalized Petersen graph $P(3k,k)$ in the projective plane

The crossing number of a graph $G$ in a surface $Σ$, denoted by $cr_Σ(G)$, is the minimum number of pairwise intersections of edges in a drawing of $G$ in $Σ$. Let $k$ be an integer satisfying $k\geq 3$, the generalized Petersen graph $P(3k,k)$ is the graph with vertex set $V(P(3k,k))=\{u_i, v_i| i=1,2,\cdots,3k\}$ and edge set $E(P(3k,k))=\{u_iu_{i+1}, u_iv_i, v_iv_{k+i}| i=1,2,\cdots,3k\},$ the subscripts are read modulo $3k.$ This paper investigates the crossing number of $P(3k,k)$ in the projective plane. We determine the exact value of $cr_{N_1}(P(3k,k))$ is $k-2$ when $3\le k\le 7,$ more…

若干循环图的分次Betti数

Graded Betti numbers of some circulant graphs

该研究计算了三类循环图边理想的$\mathbb{N}$-分次Betti数，并探讨了相关图的正则度、投射维数等代数与组合性质，以及其满足well-covered、Cohen-Macaulay等性质的条件。

Drawings of complete graphs in the projective plane

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely $n^4/64+O(n^3)$, thus matching asymptotically the conjectured value of $\text{cr}(K_n)$. Let $\text{cr}_P(G)$ denote the crossing number of a graph $G$ in the projective …

通过圆柱上的二聚体和三交叉图映射实现射影几何中的可积动力学

Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder

本文通过引入圆柱上二部图的扭曲三交叉图映射，为 Goncharov-Kenyon 簇可积系统提供几何实现，并证明五边形映射与交比动力学均为簇可积系统，最终用几何 $R$-矩阵描述交比动力学。