16941824
On the equitable chromatic number of complete n-partite graphs
Abstract In this note, we derive an explicit formula for the equitable chromatic number of a complete n-partite graph Kp1,p2,…,pn. Namely, if M is the largest integer such that p i ( mod M) p i M (i=1,2,…,n) then χ e (K p 1 ,p 2 ,…,p n )= ∑ i=1 n p i M+1 , where χe(G) is the equitable chromatic number of graph G.
3006898
A six-color theorem for the edge-face coloring of plane graphs
Abstract It was shown (Kronk and Mitchen, 1973) that the set of vertices, edges and faces of any normal map on the sphere can be colored with seven colors. In this paper we solve a somewhat different problem: the set of edges and faces of any plane graph with Δ ⩽ 3 can be colored by six colors.
24660543
A note on the lower bounds of signed domination number of a graph
Abstract Let G = (V, E) be a graph. For a function f : V → {−1, 1}, the weight of f is w(f) = Σv ∈ V f(v). For a vertex v in V, we define f[v] = Σu∈N[v] f(u). A signed dominating function of G is a function f : V → {−1, 1} such that f[v] ⩾ 1 for all v ∈ V. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. A signed dominating function of a weight γs(G) we call a γs-function of G. In this paper, we study the signed domination problem of general graph, and obtain some lower bounds of the signed domination number of a graph, and show that these lower bounds are sharp, and extend a result in Dunbar et al. (1995).
14910614
An inequality between the diameter and the inverse dual degree of a tree
Let T be a nontrivial tree with diameter D(T) and radius R(T). Let I(T) be the inverse dual degree of T which is defined to be ΣueV(T) 1/d(u), where d(u) = (Συ∈ N(u)d(υ))/d(u) for u ∈ V(T). For any longest path P of T, denote by a(P) the number of vertices outside P with degree at least 2, b(P) the number of vertices on P with degree at least 3 and distance at least 2 to each of the end-vertices of P, and c(P) the number of vertices adjacent to one of the end-vertices of P and with degree at least 3. In this note we prove that I(T) ≥ D(T)/2+ a(P)/3 + b(P)/10 + c(P)/12 + 5/6. As a corollary we then get {R(T) + 1/3 if D(T) is odd, I(T) ≥ {R(T) + 5/6 if D(T) is even, with equality if and only if T is a path of at least four vertices. The latter inequality strengthens a conjecture made by the program Graffiti.
37643061
Adjacent strong edge coloring of graphs
Abstract For a graph G ( V , E ), if a proper k -edge coloring ƒ is satisfied with C ( u ) ≠ C ( v ) for uv ∈ E ( G ), where C(u) = {ƒ(uv) | uv ∈ E} , then ƒ is called k-adjacent strong edge coloring of G , is abbreviated k-ASEC , and χ ′ as ( G ) = min { k | k - ASEC of G } is called the adjacent strong edge chromatic number of G . In this paper, we discuss some properties of χ′ as ( G ), and obtain the χ′ as ( G ) of some special graphs and present a conjecture: if G are graphs whose order of each component is at least six, then χ ′ as ( G ) ≤ Δ ( G ) + 2, where Δ( G ) is the maximum degree of G .
2737366
A proof of an inequality concerning k-restricted edge connectivity
An edge subset S of a connected graph G is a k-restricted edge cut if G-S is disconnected, and every component of G-S has at least k vertices. The k-restricted edge connectivity is the cardinality of a minimum k-restricted edge cut. In this note, we show that except for a well-defined class of graphs, k-restricted edge cuts of a connected graph G exist for any k=<@d(G)+1, where @d(G) is the minimum degree of G. Furthermore, we obtain an upper bound for k-restricted edge connectivity.
18883067
On selection of candidate paths for proportional routing
QoS routing involves selection of paths for flows based on the knowledge at network nodes about the availability of resources along paths, and the QoS requirements of flows. Several QoS routing schemes have been proposed that differ in the way they gather information about the network state and select paths using this information. Most of these schemes can be categorized as best path routing where a source node selects the "best" path for each incoming flow based on its current view of the global network state. It has been shown that best path routing schemes require frequent exchange of network state, imposing both communication overhead on the network and processing overheads on the core routers. On the other hand, proportional routing schemes proportion incoming flows among a set of candidate paths. Two key questions that arise under proportional routing are how to select candidate paths and how to proportion flows among candidate paths. We propose a scheme that selects a few widest disjoint paths as candidates and equalizes the blocking probabilities of the candidate paths. We show that our proportional routing approach yields higher throughput with lower overhead than best path routing approach. Furthermore, we present a method for aggregating the state of an area and extend the proportional routing approach to provide hierarchical routing across multiple areas in a large network.
5156351
Model-based bundle adjustment with application to face modeling
We present a new model-based bundle adjustment algorithm to recover the 3D model of a scene/object from a sequence of images with unknown motions. Instead of representing scene/object by a collection of isolated 3D features (usually points), our algorithm uses a surface controlled by a small set of parameters. Compared with previous model-based approaches, our approach has the following advantages. First instead of using the model space as a regularizer we directly use it as our search space, thus resulting in a more elegant formulation with fewer unknowns and fewer equations. Second, our algorithm automatically associates tracked points with their correct locations on the surfaces, thereby eliminating the need for a prior 2D-to-3D association. Third, regarding face modeling, we use a very small set of face metrics (meaningful deformations) to parameterize the face geometry, resulting in a smaller search space and a better posed system. Experiments with both synthetic and real data show that this new algorithm is faster, more accurate and more stable than existing ones.
38207892
On the Relations Between Arboricity and Independent Number Or Covering Number
In this paper, the following results are obtained: a(G) + β(G) ⩽ p + 1, ⌈p2⌉ ⩽ a(G)β(G), ⌈p2⌉ ⩽ a′(G) + gb′(G) ⩽ pa′(G) + α′(G) ⩽ 2⌈p2⌉ (δ ⩽ 1) and all bounds are sharp, where p = ¦V(G)¦, ⌈x⌉ denotes the smallest integer greater than or equal to x, a(G) is the vertex arboricity, a′(G) is the edge arboricity, α′(G) is the edge covering number, β(G) is the vertex independent number, β′(G) is the edge independent number, and δ is the minimum degree of G.
2290480
Optimal multicast smoothing of streaming video over an internetwork
A number of applications such as internet video broadcasts, corporate telecasts, distance learning etc. require transmission of streaming video to multiple simultaneous users across an internetwork. The high bandwidth requirements coupled with the multi-timescale burstiness of compressed video make it a challenging problem to provision network resources for transmitting streaming multimedia. For such applications to become affordable and ubiquitous, it is necessary to develop scalable techniques which can efficiently deliver streaming video to multiple heterogeneous clients across a heterogeneous internetwork. We propose using multicasting of smoothed video and differential caching of the video at intermediate nodes in the distribution tree, as techniques for reducing the network bandwidth requirements of such dissemination. We formulate the multicast smoothing problem, and develop an algorithm for computing the set of optimally smoothed transmission schedules for the tree (such that the transmission schedule along each link in the tree has the lowest peak rate and rate variability for any feasible transmission schedule for that link) given a buffer allocation to the different nodes in the tree. We also develop an algorithm to compute the minimum total buffer allocation to the entire tree and the corresponding allocation to each node, such that feasible transmission is possible to all the clients, when the tree has heterogeneous rate constraints. MPEG-2 trace-driven performance evaluations indicate that there are substantial benefits from multicast smoothing and differential caching. For example, the optimal multicast smoothing can reduce the total transmission bandwidth requirements in the distribution tree by more than a factor of 3 as compared to multicasting the unsmoothed stream.