ContributionsOn optimal orientations of cartesian products of graphs (I)

For a graph G, let D(G) be the family of strong orientations of G, and define [ovbar|d] (G) = min[d(D) vb D ] D(G), where d(D) denotes the diameter of the digraph D. Let G × H denote the cartesian product of the graphs G and H. In this paper, we determine completely the values of and , except , where K_n, P_n and C_n denote the complete graph, path and cycle of order n, respectively.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Graphs having distance-n domination number half their order

For any positive integer n and any graph G a set D of vertices of G is a distance-n dominating set, if every vertex vV(G)−D has exactly distance n to at least one vertex in D. The distance-n domination number γ_=n(G) is the smallest number of vertices in any distance-n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance-n domination number has the upper bound p/2 as Ore's upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance-n domination number equal to half their order, when the diameter is greater or equal 2n−1. With this result we confirm a conjecture of Boland, Haynes, and Lawson.     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).     

On a problem concerning ordered colourings

Let G be a connected graph with v(G) 2 vertices and independence number (G). G is critical if for any edge e of G:

1. (i) (G − e) > (G), if e is not a cut edge of G, and

2. (ii) v(G_i) − (G_i) < v(G) − (G), I = 1, 2, if e is a cut edge and G₁, G₂ are the two components of G − e.

Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then with equality possible if and only if G is a tree. In this paper we establish this conjecture.     

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T⁻, thus (T,T⁻) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T⁻, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.     

图的区间边着色的收缩图方法

图G的一个用了颜色1,2,…,t的边着色称为区间t-着色,如果所有t种颜色都被用到,并且关联于G的同一个顶点的边上的颜色是各不相同的,且这些颜色构成了一个连续的整数区间.G称作是可区间着色的,如果对某个正整数t,G有一个区间t-着色.所有可区间着色的图构成的集合记作■.对图G∈■,使得G有一个区间t-着色的t的最小值和最大值分别记作ω(G)和W(G).现给出了图的区间着色的收缩图方法.利用此方法,我们对双圈图G∈■,证明了ω(G)=△(G)或△(G)+1,并且完全确定了ω(G)=△(G)及ω(G)=△(G)+1的双圈图类.     

6-圈至多含一弦平面图的线性荫度

线性森林是指每个连通分支都是路的图.图G的线性荫度la(G)等于将其边分解为k个边不交的线性森林的最小整数k.文中利用权转移方法证明了,若G是一个最大度大于等于7且每个6-圈至多含一条弦的平面图,则la(G)=「(△(G))/2」.     

No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.     

Remarks on the bondage number of planar graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. In 1998, J.E. Dunbar, T.W. Haynes, U. Teschner, and L. Volkmann posed the conjecture b(G)Δ(G)+1 for every nontrivial connected planar graph G. Two years later, L. Kang and J. Yuan proved b(G)8 for every connected planar graph G, and therefore, they confirmed the conjecture for Δ(G)7. In this paper we show that this conjecture is valid for all connected planar graphs of girth g(G)4 and maximum degree Δ(G)5 as well as for all not 3-regular graphs of girth g(G)5. Some further related results and open problems are also presented.     

A survey of graph laplacians

Let G be a graph on n vertices. Its Laplacian is the n-by-n matrix L(G)-D(G)-A(G), where D(G) is the diagonal matrix of vertex degrees and A(G) is the (0,1)-adjacency matrix of G. This article surveys recent results on graph Laplacians.     

The generalized S-graphs of diameter 3

A graph is called a generalized S-graph if for every vertex v of G there exists exactly one vertex which is more remote from v than every vertex adjacent to v. A generalized S-graph of diameter 3 is called reducible if there is a pair of diametrical vertices v and such that v is also a generalized S-graph of diameter 3. Here we determine all irreducible generalized S-graphs of diameter 3.     

On total covers of graphs

A total cover of a graph G is a subset of V(G)E(G) which covers all elements of V(G)E(G). The total covering number ₂(G) of a graph G is the minimum cardinality of a total cover in G. In [1], it is proven that ₂(G)[n/2] for a connected graph G of order n. Here we consider the extremal case and give some properties of connected graphs which have a total covering number [n/2]. We prove that such a graph with even order has a 1-factor and such a graph with odd order is factor-critical.     

The relationship between the threshold dimension of split graphs and various dimensional parameters

Let the coboxicity of a graph G be denoted by cob(G), and the threshold dimension by t(G). For fixed k≥3, determining if cob(G)≥k and t(G)≤k are both NP-complete problems. We show that if G is a comparability graph, then we can determine if cob(G)≤2 in polynomial time. This result shows that it is possible to determine if the interval dimension of a poset equals 2 in polynomial time. If the clique covering number of G is 2, we show that one can determine if t(G)≤2 in polynomial time. Sufficient conditions on G are given for cob(G)≤2 and for t(G)≤2.     

Cyclic coloring of plane graphs

Let G be a plane graph, and let χ_k(G) be the minimum number of colors to color the vertices of G so that every two of them which lie in the boundary of the same face of the size at most k, receive different colors. In 1966, Ore and Plummer proved that χ_k(G)2k for any k3. It is also known that χ₃(G)4 (Appel and Haken, 1976) and χ₄(G)6 (Borodin, 1984). The result in the present paper is: χ₅(G)9, χ₆(G)11, χ₇(G)12, and χ_k(G)2k − 3 if k8.     

On index of total boundedness of (strictly) o-bounded groups

Let G be a metrizable topological group. Denote by itb(G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ-bounded if itb(G)₀. The group G is called o-bounded if for every sequence (U_n)_nω of neighborhoods of the identity in G there exists a sequence (F_n)_nω of finite subsets in G such that G=_nωF_n·U_n; G is called strictly o-bounded (respectively OF-determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood U_n of the identity in G and a finite subset F_n of G, respectively. The player II wins if G=_nωF_n·U_n.

For a second countable group G the following results are proven. . If G is strictly o-bounded, then itb(G)₁ and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies . G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii) ; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

On <Emphasis Type="Italic">σ</Emphasis>-semipermutable Subgroups of Finite Groups

Let σ={σ_i|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σ_i|σ_i ∩ π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member≠1 of H is a Hall σ_i-subgroup of G for some σ_i ∈ σ and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). A subgroup H of G is said to be:σ-semipermutable in G with respect to H if HH_i^x=H_i^xH for all x ∈ G and all H_i ∈ H such that (|H|,|H_i|)=1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.     

Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.