Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6)  35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

ContributionsUpper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is calledΓ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) = IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

C-normality and solvability of groups

A subgroup H is called c-normal in group G if there exists a normal subgroup N and G such that HN = G and H ∩ N ≤ H_G where H_G =: Core(H) = _gG H^g is the maximal normal subgroup of G which is contained in H. We obtain some results about the c-normal subgroups and the solvability of groups.     

Noncommutative-nilpotent matrices and the jacobian conjecture

We prove that a polynomial map F = X + H ∈ k[X] with homogeneous H(k is a field of characteristic zero) is linear triangularizable if and only if the Jacobian matrix J(H) is a noncommutative-nilpotent matrix.     

On <Emphasis Type="Italic">σ</Emphasis>-semipermutable Subgroups of Finite Groups

Let σ={σ_i|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σ_i|σ_i ∩ π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member≠1 of H is a Hall σ_i-subgroup of G for some σ_i ∈ σ and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). A subgroup H of G is said to be:σ-semipermutable in G with respect to H if HH_i^x=H_i^xH for all x ∈ G and all H_i ∈ H such that (|H|,|H_i|)=1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.