Related papers: Finite $3$-connected homogeneous graphs

This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph $K_n$ is a partition of its edges into disjoint classes. Each class of edges in a…

For two vertex disjoint graphs $H$ and $F$, we use $H\cup F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)$, and use $H+F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup…

Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $\Gamma_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

Let $G = (V,E)$ be a finite simple graph. Recall that a proper coloring of $G$ is a mapping $\varphi: V\to\{1,\ldots,k\}$ such that every color class induces an independent set. Such a $\varphi$ is called a semi-matching coloring if the…

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma(v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and for any field $F$, let…

An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if any edge of the graph is sent to be a line segment. And we say that an embedding $f$ of a graph $G$ into $\mathbb{R}^3$ is free, if $\pi_1(\mathbb{R}^3-f(G))$ is a free…

For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if they have distance at most $t$ in $G$; and the $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices are…

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…

We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.

Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the…

Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $…

We consider a distance-regular graph $\G$ with diameter $d \ge 3$ and eigenvalues $k=\theta_0>\theta_1>... >\theta_d$. We show the intersection numbers $a_1, b_1$ satisfy $$ (\theta_1 + {k \over a_1+1}) (\theta_d + {k \over a_1+1}) \ge -…

An automorphism on a graph $G$ is a bijective mapping on the vertex set $V(G)$, which preserves the relation of adjacency between any two vertices of $G$. An automorphism $g$ fixes a vertex $v$ if $g$ maps $v$ onto itself. The stabilizer of…

Given a finitely generated $G$ and a subgraph $H \leq G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{Sch}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite…

Two graphs $G$ and $H$ are homomorphism indistinguishable over a class of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Many…

Let the join of two graphs be the union of two disjoint graphs connected by $j$ edges in a one-to-one manner. In previous work by Gyurov and Pinzon, which generalized the results of Badura and Rara, the determinant of the adjacency matrix…

A countable band $B$ is called homogeneous if every isomorphism between finitely generated subbands extends to an automorphism of $B$. In this paper we give a complete classification of all the homogeneous bands. We prove that a homogeneous…