Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

No optimal 1‐planar graph triangulates any nonorientable closed surface

Suzuki [Discrete Math. 310 (2010), 6–11] proved that for any orientable closed surface F² other than the sphere, there exists an optimal 1‐planar graph which can be embedded on F² as a triangulation. However, for nonorientable closed surfaces, the existence of such graphs is unknown. In this article, we prove that no optimal 1‐planar graph triangulates a nonorientable closed surface.     

基于不同条件的旅游路线规划问题研究

运用2015年全国研究生数学建模竞赛F题的数据资料,针对旅游路线合理规划问题的第一问展开研究.以F题的问题一为起点进行了分析研究,是因为第一问的完成是解决后续问题的关键.首先通过地图搜集并补全了缺失数据并对数据进行合理的处理,然后采用将旅游年数最少目标转化为该最小生成树的"最少圈覆盖"方法来进行对问题一的求解,也可以理解为一个广义的多旅行商问题,以旅行商的人数(即年数)最少为目标.采用Dijkstra算法、最少圈覆盖法、智能算法和图论聚类等方法,通过这些方法建立了单目标优化模型,并运用旅行商问题和模型之间的转换来对问题进行分析与求解.