A homogeneous tournament is a tournament with 4 t + 3 vertices such that every arc is contained in exactly t + 1 cycles of length 3. Homogeneous tournaments are the first class of tournaments that are proved to be path extendable, which means that every nonhamiltonian path P in such a tournament T can be extended to a path P ′ with the same initial and terminal vertex and V ( P ′ ) = V ( P ) ∪ { u } for a certain vertex u ∈ V ( T ) ∖ V ( P ) . In order to find more path extendable tournaments we study the generalization of homogeneous tournaments called near-homogeneous tournaments, in which every arc is contained in t or t + 1 cycles of length 3. Near-homogeneity has been defined in tournaments with 4 t + 1 vertices. In this paper, we raise a new method to construct near-homogeneous tournaments with 4 t + 1 vertices. We then show that the definition of near-homogeneous tournament can be extended to tournaments with an even number of vertices. Finally we verify path extendability of near-homogeneous tournaments, thus expand the class of path extendable tournaments.

Cyclic vertex connectivity cκ and cyclic edge connectivity cλ are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that cκ=cλ for (4,5,6)-fullerene graphs except for four exceptional graphs with order less than 16. We also give O(ν)-algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4,5,6)-fullerene graphs.

In this paper, we consider the perfect demand matching problem ( PDM ) which combines aspects of the knapsack problem along with the b -matching problem. It is a generalization of the maximum weight matching problem which has been fundamental in the development of theory of computer science and operations research . This problem is NP-hard and there exists a constant ϵ > 0 such that the problem admits no 1 + ϵ -approximation algorithm, unless P=NP. Here, we investigate the performance of a distributed message passing algorithm called Max-sum belief propagation for computing the problem of finding the optimal perfect demand matching. As the main result, we demonstrate the rigorous theoretical analysis of the Max-sum BP algorithm for PDM , and establish that within pseudo-polynomial-time, our algorithm could converge to the optimal solution of PDM , provided that the optimal solution of its LP relaxation is unique and integral. Different from the techniques used in previous literature, our analysis is based on primal-dual complementary slackness conditions , and thus the number of iterations of the algorithm is independent of the structure of the given graph. Moreover, to the best of our knowledge, this is one of a very few instances where BP algorithm is proved correct for NP-hard problems.

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest$k$-path problem ($k$-VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph$G$of order$n$, BP algorithm converges to the unique optimal solution of$k$-VDSP on$G$within$O(n^{2}w_{max})$iterations, provided that the weight$w_{e}$is nonnegative integral for each arc$e\in E(G)$, where$w_{max}=\max \{w_{e}: e\in E(G)\}$. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of$k$-VDSP to the case of multiple sources or sinks.

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles.

In 1972, Woodall raised the following Ore-type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+(u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w) +d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is contained in a Hamilton cycle. The lower bounds in both results are tight. In this paper, we reduce both bounds by $1$ and prove that the conclusions still hold, with only a few exceptional cases that can be clearly characterized.