Indistinguishability of Percolation Clusters

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to non-decay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for $p_u$.Comment: To appear in Ann. Proba

Sliding piece puzzles with oriented tiles

AbstractIn this paper, we consider n identical tiles which are placed on the n + 1 vertices of a graph and which move along the edges of the graph. The tiles come with an “orientation”, an element of an arbitrary finite group H. Moving a tile along a given edge into the empty vertex changes the orientation of the tile in a prescribed way. We study the group of oriented positions of the tiles achievable from an initial position which fix the empty vertex. It may be thought of as a subgroup of the semidirect product Hn⋊Sn or the wreath product H wr Sn

Group Connectivity of Graphs

Tutte introduced the theory of nowhere-zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere-zero A-flow, for any Abelian group A with |A| ≥ k. In 1992 Jaeger et al. [16] extended nowhere-zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b: V (G) A with sumv∈ V(G ) b(v) = 0, there always exists a map ƒ: E(G) A - {lcub}0{rcub}, such that at each v ∈ V(G), e=vw isdirectedfrom vtow fe- e=uvi sdirectedfrom utov fe=b v in A, then G is A-connected. For a 2-edge-connected graph G, define Lambda g(G) = min{lcub}k: for any Abelian group A with |A| ≥ k, G is A-connected{rcub}.;Let G1 ⊗ G2 and G1 xG2 denote the strong and Cartesian product of two connected nontrivial graphs G1 and G2. We prove that Lambdag(G 1 ⊗ G2) ≤ 4, where equality holds if and only if both G1 and G 2 are trees and min{lcub}|V (G1)|, |V (G2)|{rcub}=2; Lambda g(G1 ⊗ G 2) ≤ 5, where equality holds if and only if both G 1 and G2 are trees and either G 1 ≅ K1, m and G2 ≅ K 1,n, for n, m ≥ 2 or min{lcub}|V (G1)|, | V (G2)|{rcub}=2. A similar result for the lexicographical product graphs is also obtained.;Let P denote a path in G, let beta G(P) be the minimum length of a circuit containing P, and let betai(G) be the maximum of betaG(P) over paths of length i in G. We show that Lambda g(G) ≤ betai( G) + 1 for any integer i \u3e 0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [J. Comb. Theory, Ser. B, 98(6) (2008), 1325-1336], among others. We completely determine all graphs G with Lambda g(G) = beta2(G) + 1.;Let Z3 denote the cyclic group of order 3. In [16], Jaeger et al. conjectured that every 5-edge-connected graph is Z3 -connected. We proved the following: (i) Every 5-edge-connected graph is Z3 -connected if and only if every 5-edge-connected line graph is Z3 -connected. (ii) Every 6-edge-connected triangular line graph is Z3 -connected. (iii) Every 7-edge-connected triangular claw-free graph is Z3 -connected. In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere-zero 3-flow

Recent developments on the power graph of finite groups - a survey

Funding: Ajay Kumar is supported by CSIR-UGC JRF, New Delhi, India, through Ref No.: 19/06/2016(i)EU-V/Roll No: 417267. Lavanya Selvaganesh is financially supported by SERB, India, through Grant No.: MTR/2018/000254 under the scheme MATRICS. T. Tamizh Chelvam is supported by CSIR Emeritus Scientist Scheme of Council of Scientific and Industrial Research (No.21 (1123)/20/EMR-II), Government of India.Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M_{11}) and subsection 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.Publisher PDFPeer reviewe