SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2424-2447, January 2020.
Any graph $G$ with chromatic number $k$ can be obtained using a Hajós-type construction, i.e., by iteratively performing the Hajós merge and vertex identification operations on a sequence of graphs starting with $K_k$. Finding such constructions for a given graph or family of graphs is a challenging task. In this paper, we show that for a large class of Hajós merges and for any identification of a pair of vertices having distance at least five from each other, the resulting graph has an $S^1$-wedge summand in its neighborhood complex. Our results imply that for a bridgeless graph $G$ with a highly connected neighborhood complex, the final step in a Hajós construction must be a vertex identification with vertices at short distance from each other. We investigate this restriction in detail. We also introduce two graph construction algorithms using Hajós-type operations. We discuss the results of computational experiments in which we investigate the rank of the first homology group of the neighborhood complexes of graphs generated using these algorithms.

中文翻译：

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Hajós型建筑和邻里建筑群

SIAM离散数学杂志，第34卷，第4期，第2424-2447页，2020年1月。
任何具有色数$ k $的图$ G $都可以使用Hajós类型的构造来获得，即，通过对以$ K_k $开头的一系列图重复执行Hajós合并和顶点识别操作。为给定图或图族找到这种构造是一项艰巨的任务。在本文中，我们表明，对于大量的Hajós合并，以及对任何距离至少为5的一对顶点的任何标识，所得图在其邻域复数中具有$ S ^ 1 $-楔形求和。我们的结果表明，对于具有高度连通的邻域复合体的无桥图$ G $，Hajós构造的最后一步必须是顶点相互之间具有短距离的顶点识别。我们将详细研究此限制。我们还介绍了两种使用Hajós型运算的图构造算法。我们讨论了计算实验的结果，其中我们调查了使用这些算法生成的图的邻域络合物的第一同源组的等级。