The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τα(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) − 2)h(G) ≥...

Let G be a simple planar graph and let α be a positive integer such that 1 ≤ α ≤ 3 and δ(G) ≥ α + 2. For every pair of edges e1, e2 of G, the graph G− {e1, e2} contains an [α, α + 1]-factor.

For α, β > 0 and for a locally integrable function (or, more generally , a distribution) ϕ on (0, ∞), we study integral ooperators G α,β ϕ on L 2 (R +) defined by G α,β ϕ f (x) = R+ ϕ x α + y β f (y)dy. We describe the bounded and compact operators G α,β ϕ and operators G α,β ϕ of Schatten–von Neumann class S p. We also study continuity properties of the averaging projection Q α,β onto the oper...

Suppose g = a semi-simple Lie algebra over C G = the corresponding adjoint group B = a Borel subgroup of G T = a Cartan subgroup of B b, t = corresponding Lie algebras Σ = roots corresponding to the choice of t ∆ = simple roots corresponding to the choice of b W = the Weyl groupNG(T )/T . For each α in ∆, let eα 6= 0 be an element of the root space gα. The triple (b, t, {eα}) is called a frame ...

Let G be a simple graph on n vertices. LetH be either the complete graph Km or the complete bipartite graph Kr,s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if βi,α(H) ≤ βi,α(G) for all i ≥ 0 and α ∈ Z. In fact, it suffices to consider only the first syzygy module. In particular, we prove that β1,α(H) ≤ β1,α(G) for all α ∈ Z if and only if G contains a ...

where σ1(α), . . . , σd(α) are all the distinct values of σ(α) as σ runs over the Galois group. For each σ ∈ G, σ(α) = σi(α) for a unique i from 1 to d. Moreover, σ(α) = σi(α) if and only if σ ∈ σiH, where H = {τ ∈ G : τ(α) = α} = Gal(L/K(α)). Therefore as σ runs over G, the number σi(α) appears as σ(α) whenever σ is in the left coset σiH, so σi(α) occurs |H| times, and |H| = [L : K(α)] = [L : ...

Let α(G) be the stability number of a graph G. A class of graphs P is called α-polynomial if there exists a polynomial-time algorithm to determine α(G) for G ∈ P. For every hereditary α-polynomial class P we construct a hereditary extension of P which is either an α-polynomial class or α can be approximated in polynomial time in the extended class.

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian one exception of family graphs. If a nontrivial path contains only internal vertices degree two and end not two, then we call it branch. A set S branches called branch cut if delete all edges leading to more components than G. We use bond denote minimal cut. branch-bond has an o...

For a graphG = (V,E), let bc(G) denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G, bc(G) ≤ n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80s that for almost every graph G equality holds, i.e., that for the random graph ...