Connectivity Properties of Generalized K4-Hypercubes

We consider the class of generalized hypercubes constructed recursively from the graph [Formula: see text] by repeatedly taking two copies of such a graph with a perfect matching added in between. We show that all graphs obtained this way have very good connectivity properties. They are all maximally connected, and even when linearly many vertices are deleted, the remaining graph will have a large connected component with only a few vertices in other components. We also show examples that we can delete more vertices in certain graphs in this class to get the second largest component to have certain sizes, including the case when we get two components of equal size. We conjecture that these examples are best possible.

Note on Applications of Linearly Many Faults

Most graphs have this property: after removing a linear number of vertices from a graph, the surviving graph is either connected or consists of a large connected component and small components containing a small number of vertices. This property can be applied to derive fault-tolerance related network parameters: extra edge connectivity and component edge connectivity. Using this general property, we obtained the $h$-extra edge connectivity and $(h+2)$-component edge connectivity of augmented cubes, Cayley graphs generated by transposition trees, complete cubic networks (including hierarchical cubic networks), generalized exchanged hypercubes (including exchanged hypercubes) and dual-cube-like graphs (including dual cubes).

Client-Waiter Games on Complete and Random Graphs

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

Connectivity Results of Complete Cubic Networks as Associated with Linearly Many Faults

We propose the complete cubic network structure to extend the existing class of hierarchical cubic networks, and establish a general connectivity result which states that the surviving graph of a complete cubic network, when a linear number of vertices are removed, consists of a large (connected) component and a number of smaller components which altogether contain a limited number of vertices. As applications, we characterize several fault-tolerance properties for the complete cubic network, including its restricted connectivity, i.e., the size of a minimum vertex cut such that the degree of every vertex in the surviving graph has a guaranteed lower bound; its cyclic vertex-connectivity, i.e., the size of a minimum vertex cut such that at least two components in the surviving graph contain a cycle; its component connectivity, i.e., the size of a minimum vertex cut whose removal leads to a certain number of components in its surviving graph; and its conditional diagnosability, i.e., the maximum number of faulty vertices that can be detected via a self-diagnostic process, in terms of the common Comparison Diagnosis model.

Component Games on Regular Graphs

We study the (1:b) Maker–Breaker component game, played on the edge set of ad-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's biasb, we determine whether Breaker wins (on anyd-regular graph) or Maker wins (on almost everyd-regular graph) and provide explicit winning strategies for both players.To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.