We introduce two variations of the cops and robber game on graphs. These games yield invariants in Z+∪{∞} for any connected graph Γ, weak cop number wCop(Γ) strong sCop(Γ). satisfy that sCop(Γ)≤wCop(Γ). Any is finite or a tree has one. new are preserved under small local perturbations graph, specifically, both numbers quasi-isometric More generally, we prove if Δ quasi-retract Γ then wCop(Δ)≤wC...

The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell [6]. The game on a graph G can be described as follows: let fn be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time n ≥ 1, fn vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices o...

Let ? be a finite, undirected, connected, simple graph. We say that matching M is permutable m -matching if contains edges and the subgroup of Aut ( ) fixes setwise allows to permuted in any fashion. A 2-transitive stabilizer can map ordered pair distinct other . provide constructions graphs with matching; we show that, an arc-transitive graph for ? 4 , then degree at least ; and, when sufficie...

Abstract A graph is said to be ‐ ubiquitous , where the minor relation between graphs, if whenever a with for all then one also has disjoint union of many copies . well‐known conjecture Andreae that every locally finite connected ‐ubiquitous. In this paper we give sufficient condition on structure ends which implies particular full‐grid

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

a r t i c l e i n f o a b s t r a c t A graph Γ is k-CS-transitive, for a positive integer k, if for any two connected isomorphic induced subgraphs A and B of Γ , each of size k, there is an automorphism of Γ taking A to B. The graph is called k-CS-homogeneous if any isomorphism between two connected induced subgraphs of size k extends to an automor-phism. We consider locally-finite infinite k-...

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sp...