Yang Yuansheng

Maximizing battery life routing in wireless ad hoc networks

Most wireless ad hoc networks consist of mobile devices which operate on batteries. Power consumption in this type of network therefore is paramount important. To maximize the lifetime of an ad hoc network, it is essential to prolong each individual node (mobile) life through minimizing the total transmission energy consumption for each communication request. Therefore, an efficient routing protocol must satisfy that the energy consumption rate at each node is evenly distributed and at the same time the total transmission energy for each request is minimized. To devise a routing protocol meeting the above two conflict objectives simultaneously is very difficult due to the NP hardness of the problem. Instead, an approximation solution is desirable and appreciated. In this paper, we focus on developing power-aware routing algorithms which find routing paths that maximize the lifetime of individual nodes and minimize the total transmission energy consumption, thereby prolong the life of the entire network. In particular, for an ad hoc network consisting of the same type of battery mobile nodes, two approximation algorithms are proposed. The running times of the proposed algorithms are determined by the accuracies of the approximation solutions. Compared with a previously known result, our algorithms have less energy overhead and can be implemented in a distributed environment. For an ad hoc network equipped with different types of battery mobile nodes, a new power-aware routing protocol is proposed, and the algorithm can also be implemented in a distributed manner.

Roman domination in regular graphs

A Roman domination function on a graph G = (V(G), E(G)) is a function f : V(G)→ {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V(G)) = ∑ u∈V(G) f (u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11–22] showed that γ(G) ≤ γR(G) ≤ 2γ(G) and defined a graph G to be Roman if γR(G) = 2γ(G). In this article, the authors gave several classes of Roman graphs: P3k, P3k+2, C3k, C3k+2 for k ≥ 1, Km,n for min{m, n} 6= 2, and any graph G with γ(G) = 1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs C(n; {1, 3})(n ≥ 7, n 6≡ 4 (mod 5)) and C(n; {1, 2, . . . , k}) (k ≤ b n 2 c), n 6≡ 1 (mod (2k+ 1)), (n 6= 2k) are Roman graphs, (2) the generalized Petersen graphs P(n, 2k + 1)(n 6= 4k + 2, n ≡ 0 (mod 4) and 0 ≤ k ≤ b n 2 c), P(n, 1) (n 6≡ 2 (mod 4)), P(n, 3) (n ≥ 7, n 6≡ 3 (mod 4)) and P(11, 3) are Roman graphs, and (3) the Cartesian product graphs C5m C5n(m ≥ 1, n ≥ 1) are Roman graphs.

The minimum number of vertices with girth 6 and degree set D = {r,m}

A (D;g)-cage is a graph having the minimum number of vertices, with degree set D and girth g. Denote by f(D;g) the number of vertices in a (D;g)-cage. In this paper it is shown that f({r,m}; 6) ≥ 2(rm - m + 1) for any 2 ≤ r < m, and f({r, m}; 6) = 2(rm - m + 1) if either (i) 2 ≤ r ≤ 5 and r < m or (ii) m - 1 is a prime power and 2 ≤ r < m. Upon these results, it is conjectured that f({r, m}; 6) = 2(rm - m + 1) for any r with 2 ≤ r < m.

Equitable total coloring of Cm Cn

The equitable total chromatic number of a graph G is the smallest integer k for which G has a k-total coloring such that the number of vertices and edges colored with each color differs by at most one. In this paper, we show that the Cartesian product graphs of Cm and Cn have equitable total 5-coloring for all m>=3 and n>=3.

Domination dot-critical graphs with no critical vertices

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. Burton and Sumner [Discrete Math. 306 (2006) 11–18] posed the problem: Is it true that for k 4, there exists a totally k-dot-critical graph with no critical vertices? In this paper, we show that this problem has a positive answer.

The planar Ramsey number PR(C4,K7)☆

The planar Ramsey number PR(H1, H2) is the smallest integer n such that any planar graph on n vertices contains a copy of H1 or its complement contains a copy of H2. It is known that the Ramsey number R(C4, K7) = 22. The planar Ramsey numbers PR(C4, Kl ) for l ≤ 6 are known. In this paper we show that PR(C4, K7) = 20. c

The planar Ramsey number PR(K4-e,K5)☆

The planar Ramsey number PR(H1,H2)PR(H1,H2) is the smallest integer n such that any planar graph on n vertices contains a copy of H1H1 or its complement contains a copy of H2H2. It is known that the Ramsey number R(K4-e,K5)=16R(K4-e,K5)=16. The planar Ramsey numbers PR(K4-e,K3)=7PR(K4-e,K3)=7 and PR(K4-e,K4)=10PR(K4-e,K4)=10 are known. In this paper we show that PR(K4-e,K5)=14PR(K4-e,K5)=14.

The Ramsey number r ( K 1 + C 4 , K 5 − e )

The graph K5 - e is obtained from the complete graph K5 by deleting one edge, while K1 + C4 is obtained from K5 by deleting two independent edges. With the help of a computer it is shown that r(K1 + C4, K5 - e) = 17.

On the crossing numbers of Km□Cn and Km,l□Pn

A lubricating oil composition designed for use in medium and high speed marine diesel engine crankcases which has a Total Base Number from about 5 to 40 and contains a mineral lubricating oil, an overbased calcium sulfonate, an overbased sulfurized calcium phenate, a zinc dihydrocarbyl dithiophosphate, an alkenylsuccinimide, and a friction reducing amount of at least one acyl glycine oxazoline derivative.

Note: The planar Ramsey number PR(K4-e, K5)

The planar Ramsey number PR(H1,H2)PR(H1,H2) is the smallest integer n such that any planar graph on n vertices contains a copy of H1H1 or its complement contains a copy of H2H2. It is known that the Ramsey number R(K4-e,K5)=16R(K4-e,K5)=16. The planar Ramsey numbers PR(K4-e,K3)=7PR(K4-e,K3)=7 and PR(K4-e,K4)=10PR(K4-e,K4)=10 are known. In this paper we show that PR(K4-e,K5)=14PR(K4-e,K5)=14.