Rings whose total graphs have genus at most one

Abstract

Let R be a commutative ring with Z(R) its set of zero-divisors. In this paper, we study the total graph of R, denoted by T(Γ(R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We investigate properties of the total graph of R and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer g, there are only finitely many finite rings whose total graph has genus g.2000 Mathematics Subject Classification. 05C75, 13A15.

DOMINATION IN THE TOTAL GRAPH ON ℤn

“…For any composite integer n > 1, the total graph T Γ (Z n ) contains no isolated vertex and no vertex of degree n − 1. Now we apply a lemma proved by Maimani et al, [11,Lemma 1.1] to the ring Z n and state the particular case, which is very much useful for discussions in the sequel.…”

The Total Graphs of Finite Rings

“…In [1] Akbari et al proved that if the total graph of a finite commutative ring is connected then it is also a Hamiltonian graph. In [6], Maimani et al gave the necessary and sufficient conditions for the total graphs of finite commutative rings to be planar or toroidal and in [8] Tamizh Chelvam and Asir characterized all commutative rings such that their total graphs have genus 2.…”

Finite commutative rings with higher genus unit graphs

“…Genus two zero divisor graphs of local rings were investigated by Bloomfield and Wickham in [5]. Recently, Maimani et al [14] determined all isomorphism classes of finite rings whose total graphs have genus at most one, and Tamizh Chelvam and Asir [17] characterized all isomorphism classes of finite rings whose total graphs have genus two. For a finite ring R, the unit graph G(R) is the complement of the total graph of the ring R. In [3, Theorem 5.14], all finite rings having planar unit graphs are completely classified, and in [7] toroidal ones are completely determined.…”

“…The others are (14,27) or (14,57) or (24, 57). If (14,27) is, we can get a contradiction by searching neighbors of 5, 7 and 8. If (14, 57) is (respectively, (24, 57)), we can add the edge 57 in the face 48765.…”

“…The others are (14,26), (14,27) (15,24), (14,27), (15,27), (14,57) or (24, 57). If (14,27) is, we can get a contradiction by noticing neighbors of 5. If (15,24) is (respectively, (15,27 All cases result in a contradiction.…”