The projective 3-annihilating-ideal hypergraphs

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph [Formula: see text] of a commutative ring [Formula: see text] and the topological embedding of [Formula: see text] to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings [Formula: see text] (up to isomorphism) such that [Formula: see text] is a projective graph.

Locally projective graphs and their densely embedded subgraphs

The article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph $$\Gamma $$ Γ of type (n, 3) for $$n \ge 3$$ n ≥ 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam $$G_3^1\cong \{S_4 \times 2,S_4 \times 2\}$$ G 3 1 ≅ { S 4 × 2 , S 4 × 2 } having $$S_6$$ S 6 as another completion, (b) for a vertex x of $$\Gamma $$ Γ the group $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) which stabilizes every line passing through x induces on the neighbourhood $$\Gamma (x)$$ Γ ( x ) of x the (dual) natural module $$2^n$$ 2 n of $$G(x)/G_{\frac{1}{2}}(x) \cong L_n(2)$$ G ( x ) / G 1 2 ( x ) ≅ L n ( 2 ) , (c) G(x) splits over $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) , (d) the vertex-wise stabilizer $$G_1(x)$$ G 1 ( x ) of the neighbourhood of x is a non-trivial group, and (e) $$n \ne 4$$ n ≠ 4 .

On projective intersection graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

The Crossing Number of a Projective Graph is Quadratic in the Face–Width

We show that for each integer $g\geq0$ there is a constant $c_g > 0$ such that every graph that embeds in the projective plane with sufficiently large face–width $r$ has crossing number at least $c_g r^2$ in the orientable surface $\Sigma_g$ of genus $g$. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective graphs with bounded degree.

Strongly Projective Graphs

We introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph G does not admit a homomorphism to K, where K is an odd cycle or a complete graph on at least 3 vertices, then the graph G × Ks admits, up to automorphisms of K, exactly s homomorphisms to K.