We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free, edge-transitive, twin-free, finite graphs the non-abelian composition factors of the automorphism group have bounded order. We use this to show that the automorphism groups of finite graphs of bounded Hadwiger number are obtained by repeated group extensions using abelian groups, symmetric groups and groups of bounded order.
University of Ottawa（渥太华大学）；Laboratoire G-SCOP (CNRS, Univ. Grenoble Alpes)（G-SCOP实验室（法国国家科学研究中心，格勒诺布尔阿尔卑斯大学））；Carleton University（卡尔顿大学）；Monash University（莫纳什大学）
Hadwiger and Hajós conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for $H$-immersion free gr…
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is an asymmetric colouring with $2$ colours. We make progress on this conjecture in the special case of graphs with bounded maximal degree. More precisely, we prove that if every automorphism of a connected graph with maximal degree $Δ$ moves infinitely many vertices, then there is an asymmetric colouring using $\mathcal O(\sqrt Δ\log Δ)$ colours. This is the …
Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained …
On the automorphism groups of distance-regular graphs and rank-4 primitive coherent configurations
置换群$G$的最小度是不被$G$的非单位元素固定的点的最小数目。最小度的下界对$G$有很强的结构影响。在2014年，Babai证明了具有$n$顶点的强正则图的自同构群具有最小度$\geq c n$，除了已知的例外。强正则图对应于秩为3的本原凝聚构形。我们将Babai的结果推广到秩为4的本原相干构形。我们还证明了这一结果推广到直径有界的非几何距离正则图。证明结合结构和谱方法。
图$G$的区别数$D(G)$是对图$G$的顶点着色所需的最小颜色数，使得唯一的颜色保持自同构是单位元。对于无限图，$D(G)$是有界的价的上确界，而对于有限图，$Δ(G)+1$，其中$Δ(G)$是最大价。给定一个有限或无限树$T$，其有界有限价$k$和一个整数$c$，其中$2 \leq c \leq k$，我们感兴趣的是用$c$颜色对$T$的顶点着色，使得每个保色自同构固定尽可能多的顶点.在这个意义上，我们表明，总是存在一个$c$染色，所有的顶点，其距离下一个叶至少是$\lceil\log_ck\rceil$是固定的任何颜色保持自同构，在许多情况下，可以做得更好。
