Related papers: Graphs with given automorphism group and large cli…
We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.
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,…
We prove that, to every abstract group $G$, we can associate a sequence of graphs $\Gamma_n$ such that the automorphism group of $\Gamma_n$ is isomorphic to $G$ and the genus of $\Gamma_n$ is an unbounded function of $n$.
We present an infinite sequence of finite graphs with trivial automorphism group and non-trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are…
A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.
Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that $\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at most finitely many…
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\Gamma$, there are uncountably many maximal subgroups of…
The clique-width is known to be unbounded in the class of unit interval graphs. In this paper, we show that this is a minimal hereditary class of unbounded clique-width, i.e., in every hereditary subclass of unit interval graphs the…
We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands,…
The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique-width if and only if it does not contain all quasi-threshold graphs or their complements. The proof…
The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the…
The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph…
Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of…
Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge…
Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general…
In this paper we determine automorphism groups of cyclic algebraic curves defined over finite fields of any characteristic.
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…
We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $\chi$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially…
We define a notion of quantum automorphism group of Graph C*-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of underlying…
We analyze the Brill-Noether theory of trivalent graphs and multigraphs having largest possible automorphism group in a fixed genus. For trivalent multigraphs with loops of genus at least 3, we show that there exists a graph with maximal…