Search

There are some attacks on the network, such as botnet attack, DDoS attack and Local Area Network Denial attack, which are attacked on certain group of clustered nodes in the network. At present, the existing connectivity has certain defects in reflecting the fault-tolerant ability of the network under these network attacks. In order to measure the fault tolerance and reliability of a network which is attacked on certain group of clustered nodes in the network by attacker, Lin et al. (IEEE Trans Comput 70:1719–1731, 2021) proposed the g-faulty-block connectivity. A subset F ⊆ V (G) is called a g-faulty-block of a graph G if G - F is disconnected, each component of it has at least g + 1 vertices and the subgraph induced by F is connected. The cardinality of a minimum g-faulty-block of G, denoted by FB κ g (G) , is the g-faulty-block connectivity of G. Larger h-fault block connectivity means that an attacker must launch an attack on a larger block of connected nodes so that each remaining component is not too small, which in turn limits the size of the larger components. The larger the h-fault block, the more difficult it is for an attacker to accomplish this goal. In this paper, we obtain FB κ 0 (FQ n) = 2 n + 1 , FB κ 1 (FQ n) = 3 n - 1 and FB κ g (FQ n) = (g + 2) n - 3 g + 3 for 2 ≤ g ≤ n - 4 and n ≥ 7 , where FQ n is n-dimension folded hypercube. [ABSTRACT FROM AUTHOR]

The dominated coloring (dom-coloring) of a graph G is a proper coloring such that each color class is dominated by at least one vertex. The dominated chromatic number (dom-chromatic number) of G is the minimum number of color classes among all dominated colorings of G, denoted by χ dom (G) . In this paper, we study the dominated coloring of Cartesian product, direct product, lexicographic product and strong product of some graphs. [ABSTRACT FROM AUTHOR]

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographic, strong, Cartesian, and direct products and present exact values or upper bounds for these products of graphs. [ABSTRACT FROM AUTHOR]