Let α(G), α(G), κ(G) and κ (G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. We determine the finite graph families F1 and F2 such that each of the following holds. (i) If a connected graph G satisfies κ (G) ≥ α(G) − 1, then G has a spanning closed trail if and only if G is not contractible to a member of F1. (ii) If κ (G) ≥ ...

For any permutation α on the vertex set of a graph G, the permutation graph Pα(G) is obtained by taking two copies of G, denoted G and G ′, and joining u ∈ V (G) and v ∈ V (G′) if and only if α(u) = v. Let γ(G) be the domination number of G. It has been proven that for all permutations α on any graph G, γ(G) ≤ γ(Pα(G)) ≤ 2γ(G). In 1999, Dr. Weizhen Gu posited the following conjecture: For any p...

In this paper we deal with the relation lim α lim α X = lim α X for a subset X of G, where G is an -group and α is a sequential convergence on G.

Let Q2m be the generalized quaternion group of order 2 m and DN the dihedral group of order 2N . We classify the orbits in Q2m and D n pm (p prime) under the Hurwitz action. 1 The Hurwitz Action Let G be a group. For a, b ∈ G, let a = bab and a = bab. The Hurwitz action on G (n ≥ 2) is an action of the n-string braid group Bn on G . Recall that Bn is given by the presentation Bn = 〈σ1, . . . , ...

Let G = (V,E) be an undirected graph without loops and multiple edges [1], V (G) and E(G) be the sets of vertices and edges of G, respectively. The degree of a vertex x ∈ V (G) is denoted by dG(x), the maximum degree of a vertex of G-by ∆(G), and the chromatic index [2] of G-by χ(G). A graph is regular, if all its vertices have the same degree. If α is a proper edge colouring of the graph G [3]...

Chevalley’s theorem states that for any simple finite dimensional Lie algebra g: (1) the restriction homomorphism of the algebra of polynomials S(g∗) −→ S(h∗) onto the Cartan subalgebra h induces an isomorphism S(g∗)g ∼= S(h∗)W , where W is the Weyl group of g; (2) each g-invariant polynomial is a linear combination of the polynomials trρ(x)k , where ρ is a finite dimensional representation of ...

Let G be a graph. A set S of vertices of G is called a total dominating set of G if every vertex of G is adjacent to at least one vertex in S. The total domination number γt(G) and the matching number α ′(G) of G are the cardinalities of the minimum total dominating set and the maximum matching of G, respectively. In this paper, we will introduce an upper bound of the difference between γt(G) a...

We show that for each α > 0 every sufficiently large oriented graph G with δ(G), δ−(G) ≥ 3|G|/8 + α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G) + δ(G) + δ−(G) ≥ 3|G|/2 + α|G|. Up to the term α|G| this confirms a conjecture of Häggkvist [10]. We also prove an Ore-type th...

Definition 1.2. Let n ≥ 1 and let G be an abelian group. The fundamental class of K(G, n) is the cohomology class ιn ∈ H (K(G, n);G) corresponding to idG via the isomorphism H n (K(G, n);G) ∼= HomZ(G,G). More explicitly, let ψ : πnK(G, n) ∼= −→ G be some chosen identification, and let h : πn (K(G, n)) ∼= −→ Hn (K(G, n);Z) denote the Hurewicz morphism, defined by h(α) = α∗(un), where un ∈ Hn(S) ...

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree ∆ and independence number α(G), α(G) ≤ |V | 2 . It is known that α(G) < 3∆−2 5∆−2 |V |. In this paper we improve this bound when ∆ ≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α(G) ≤ 3∆−3 5∆−3 |V | when ∆ ≥ 5 and n2 ≤ 2(∆− 1).