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This paper studies the problem of under what conditions the general nonlinear networks with directional competition and perturbations to achieve fully distributed BS (bipartite synchronization). For the norm bounded disturbances with unknown upper bound, under the assumption that the signed digraph is structurally balanced and contains a directed spanning tree, a fully distributed observer-based adaptive protocols is designed for BS problem of general nonlinear networks with no leader and neural networks approximation. For the disturbances generated by exosystems, based on a disturbance observer, a fully distributed disturbance observerbased adaptive protocol is proposed to make network achieve bounded BS. In order to reduce the chattering phenomenon, some adaptive gains and adaptive parameter vector of neural networks approximation are proposed. Finally, the theoretical results are verified by two numerical simulation examples. [ABSTRACT FROM AUTHOR]

Suppose G is a simple graph. If f is a mapping from V(G) ∪ E(G) to {1,2,...,k} such that f(e), f(u), f(v) are distinct for each edge e = uv of G, then f is called an E-total coloring of G using k colors (k-E-total coloring of G, in brief). For an E-total coloring f of a graph G and any vertex x of G, we denote the set {f(x)} ∪ {f(e)|e ∈ E(G) and e is incident with x} by C(x) and refer to it as the color set of x under f. If C(u) ≠ C(v) for any two different vertices u and v of V (G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. Let χvte(G) = {k|G has a VDET coloring using k colors }. Then the positive integer χvte(G) is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K5,n is discussed in this paper and the VDET chromatic number of K5,n has been obtained. [ABSTRACT FROM AUTHOR]

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C (x) denote the set of colors of vertex x and of the edges incident with x, we call C (x) the color set of x. If C (u) = C (v) for any two different vertices u and v of V (G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by xeVt (G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K7,n (n ≥ 978) is discussed in this paper and the VDET chromatic number of K7,n (n ≥ 978) has been obtained. [ABSTRACT FROM AUTHOR]

A linear k-forest is a graph whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, it is obtained that la4(Kn,n) =[5n/8] for n ≡ 0( mod 5). [ABSTRACT FROM AUTHOR]