A study on cyclic bandwidth sum

Suppose G is a graph of p vertices. A proper labeling f of G is a one-to-one mapping f:V(G)→{1,2,…,p}. The cyclic bandwidth sum of G with respect to f is defined by CBS _f (G)=∑ _uv∈E(G)|f(v)−f(u)| _p , where |x| _p =min {|x|,p−|x|}. The cyclic bandwidth sum of G is defined by CBS(G)=min {CBS _f (G): f is a proper labeling of G}. The bandwidth sum of G with respect to f is defined by BS _f (G)=∑ _uv∈E(G)|f(v)−f(u)|. The bandwidth sum of G is defined by BS(G)=min {BS _f (G): f is a proper labeling of G}. In this paper, we give a necessary and sufficient condition for BS(G)=CBS(G), and use this to show that BS(T)=CBS(T) when T is a tree. We also find cyclic bandwidth sums of complete bipartite graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Supported in part by the National Science Council under grants NSC91-2115-M-156-001.     

The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups

A k-L(2,1)-labelling of a graph G is a mapping f:V(G)→{0,1,2,…,k} such that |f(u)?f(v)|≥2 if uv∈E(G) and f(u)≠f(v) if u,v are distance two apart. The smallest positive integer k such that G admits a k-L(2,1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are called brick product graphs or honeycomb toroidal graphs. We prove that the λ-number of such a graph is between 5 and 7, and moreover we give a characterisation of such graphs with λ-number 5.     

Integer-magic spectra of sun graphs

Let A be a non-trivial Abelian group. A graph G=(V,E) is A-magic if there exists a labeling f:E→A∖{0} such that the induced vertex set labeling f ⁺:V→A, defined by f ⁺(v)=∑f(uv) where the sum is over all uv∈E, is a constant map. The integer-magic spectrum of a graph G is the set IM(G)={k∈ℕ∣G is ℤ _k -magic}. A sun graph is obtained from an n-cycle, by attaching paths to each pair of adjacent vertices in the cycle. In this paper, we investigate the integer-magic spectra of some sun graphs. Dedicated to Prof. Frank K. Hwang, on the occasion of his 65th birthday. Supported by Faculty Research Grant, Hong Kong Baptist University.     

L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ _j,k -number of G, denoted by λ _j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| _m ≥j if u and v are adjacent; and |f(u)−f(v)| _m ≥k if u and v are at distance two apart, where |x| _m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ _j,k -number of G and denoted by σ _j,k (G). The λ _j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ _j,k -numbers of direct products of two complete graphs and the σ _j,k -numbers of direct products and Cartesian products of two complete graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work is partially supported by FRG, Hong Kong Baptist University, Hong Kong; NSFC, China, grant 10171013; and Southeast University Science Foundation grant XJ0607230.     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ k\sum_{i=1}^{d}f_{i}(v)\le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{\{k\}}(G). Note that d_t^{1}(G)d_{t}^{\{1\}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

On total weight choosability of graphs

For a graph G with vertex set V and edge set E, a (k,k′)-total list assignment L of G assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights. If for any (k,k′)-total list assignment L of G, there exists a mapping f:V∪E→? such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ _y∈N(u) f(uy)+f(u)≠∑ _x∈N(v) f(vx)+f(v), then G is (k,k′)-total weight choosable. It is conjectured by Wong and Zhu that every graph is (2,2)-total weight choosable, and every graph with no isolated edges is (1,3)-total weight choosable. In this paper, it is proven that a graph G obtained from any loopless graph H by subdividing each edge with at least one vertex is (1,3)-total weight choosable and (2,2)-total weight choosable. It is shown that s-degenerate graphs (with s≥2) are (1,2s)-total weight choosable. Hence planar graphs are (1,10)-total weight choosable, and outerplanar graphs are (1,4)-total weight choosable. We also give a combinatorial proof that wheels are (2,2)-total weight choosable, as well as (1,3)-total weight choosable.     

Multiple L(j,1)-labeling of the triangular lattice

Let n,j,k be nonnegative integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a?b|≥j if uv∈E(G), and |a?b|≥k if u and v are distance two apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G. Let n,j,k and m be nonnegative integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m?1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a?b|,m?|a?b|}≥j if uv∈E(G), and min{|a?b|,m?|a?b|}≥k if u and v are distance two apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G. This paper provides upper and lower bounds for the n-fold L(j,1)-labeling number and the n-fold circular L(j,1)-labeling number of the triangular lattice and determines the n-fold L(2,1)-labeling number and n-fold circular L(2,1)-labeling number of the triangular lattice for n≥3.     

Circular L(j,k)-labeling number of direct product of path and cycle

Let j, k and m be positive numbers, a circular m-L(j,k)-labeling of a graph G is a function f:V(G)→[0,m) such that |f(u)?f(v)| _m ≥j if u and v are adjacent, and |f(u)?f(v)| _m ≥k if u and v are at distance two, where |a?b| _m =min{|a?b|,m?|a?b|}. The minimum m such that there exist a circular m-L(j,k)-labeling of G is called the circular L(j,k)-labeling number of G and is denoted by σ _j,k (G). In this paper, for any two positive numbers j and k with j≤k, we give some results about the circular L(j,k)-labeling number of direct product of path and cycle.     

Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S _v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D _v (G,c) the minimum number of integers which must be added to S _v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ _v∈V D _v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

Packing cycles exactly in polynomial time

Let G=(V,E) be an undirected graph in which every vertex v∈V is assigned a nonnegative integer w(v). A w-packing is a collection of cycles (repetition allowed) in G such that every v∈V is contained at most w(v) times by the members of . Let 〈w〉=2|V|+∑ _v∈V ⌈log (w(v)+1)⌉ denote the binary encoding length (input size) of the vector (w(v): v∈V) ^T . We present an efficient algorithm which finds in O(|V|⁸〈w〉²+|V|¹⁴) time a w-packing of maximum cardinality in G provided packing and covering cycles in G satisfy the ℤ₊-max-flow min-cut property.     

Some results on the injective chromatic number of?graphs

A k-coloring of a graph G=(V,E) is a mapping c:V??{1,2,??,k}. The coloring c is injective if, for every vertex v??V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ?? _i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K ₄-minor free graph G with maximum degree ????1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.     

On cubic 2-independent Hamiltonian connected graphs

A graph G=(V,E) is Hamiltonian connected if there exists a Hamiltonian path between any two vertices in G. Let P ₁=(u ₁,u ₂,…,u _|V|) and P ₂=(v ₁,v ₂,…,v _|V|) be any two Hamiltonian paths of G. We say that P ₁ and P ₂ are independent if u ₁=v ₁,u _|V|=v _|V|, and u _i ≠v _i for 1<i<|V|. A cubic graph G is 2-independent Hamiltonian connected if there are two independent Hamiltonian paths between any two different vertices of G. A graph G is 1-Hamiltonian if G−F is Hamiltonian for any F⊆V∪E with |F|=1. A graph G is super 3^*-connected if there exist i internal disjoint paths spanning G for i=1,2,3. It is proved that every super 3^*-connected graph is 1-Hamiltonian. In this paper, we prove that every cubic 2-independent Hamiltonian connected graph is also 1-Hamiltonian. We present some cubic 2-independent Hamiltonian connected graphs that are super 3^*-connected, some cubic 2-independent Hamiltonian connected graphs that are not super 3^*-connected, some super 3^*-connected graphs that are not 2-independent Hamiltonian connected, and some cubic 1-Hamiltonian graphs that are Hamiltonian connected but neither 2-independent Hamiltonian connected nor super 3^*-connected. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-233-011.     

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ?-colourings of G is connected and has diameter O(|V|²), for all ?≥k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|²).     

Group testing in graphs

This paper studies the group testing problem in graphs as follows. Given a graph G=(V,E), determine the minimum number t(G) such that t(G) tests are sufficient to identify an unknown edge e with each test specifies a subset X⊆V and answers whether the unknown edge e is in G[X] or not. Damaschke proved that ⌈log ₂ e(G)⌉≤t(G)≤⌈log ₂ e(G)⌉+1 for any graph G, where e(G) is the number of edges of G. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. Later, they proved that the conjecture is true for complete bipartite graphs. Chang and Juan verified the conjecture for bipartite graphs G with e(G)≤2⁴ or for k≥5. This paper proves the conjecture for bipartite graphs G with e(G)≤2⁵ or for k≥6. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. J.S.-t.J. is supported in part by the National Science Council under grant NSC89-2218-E-260-013. G.J.C. is supported in part by the National Science Council under grant NSC93-2213-E002-28. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

Approximating the chromatic index of multigraphs

It is well known that if G is a multigraph then χ′(G)≥χ′^*(G):=max {Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′^*(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max {2|E(G[U])|/(|U|−1):U⊆V(G),|U|≥3, |U| is odd}. The conjecture that χ′(G)≤max {Δ(G)+1,⌈Γ(G)⌉} was made independently by Goldberg (Discret. Anal. 23:3–7, 1973), Anderson (Math. Scand. 40:161–175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423–460, 1979). Using a probabilistic argument Kahn showed that for any c>0 there exists D>0 such that χ′(G)≤χ′^*(G)+c χ′^*(G) when χ′^*(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>⌊(11Δ(G)+8)/10⌋; and Scheide recently improved this bound to χ′(G)>⌊(15Δ(G)+12)/14⌋. We prove this conjecture for multigraphs G with $\chi'(G)>\lfloor\Delta(G)+\sqrt{\Delta(G)/2}\rfloor$\chi'(G)>\lfloor\Delta(G)+\sqrt{\Delta(G)/2}\rfloor , improving the above mentioned results. As a consequence, for multigraphs G with $\chi'(G)>\Delta(G)+\sqrt {\Delta(G)/2}$\chi'(G)>\Delta(G)+\sqrt {\Delta(G)/2} the answer to a 1964 problem of Vizing is in the affirmative.     

On the equitable k *-laceability of hypercubes

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container C _k (u,v). A k-container is a k ^*-container if it spans all vertices of G. We define G to be a k ^*-laceable graph if there is a k ^*-container joining any two vertices from different partite sets. A k ^*-container C _k ^*(u,v)={P ₁,…,P _k } is equitable if ||V(P _i )|−|V(P _j )||≤2 for all 1≤i,j≤k. A graph is equitably k ^*-laceable if there is an equitable k ^*-container joining any two vertices in different partite sets. Let Q _n be the n-dimensional hypercube. In this paper, we prove that the hypercube Q _n is equitably k ^*-laceable for all k≤n−4 and n≥5. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. The work of H.-M. Huang was supported in part by the National Science Council of the Republic of China under NSC94-2115-M008-013.     

Upper paired-domination in claw-free graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γ_pr(G). We establish bounds on Γ_pr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree δ. We show that Γ_pr(G)≤4n/5 if δ=1 and n≥3, Γ_pr(G)≤3n/4 if δ=2 and n≥6, and Γ_pr(G)≤2n/3 if δ≥3. All these bounds are sharp. Further, if n≥6 the graphs G achieving the bound Γ_pr(G)=4n/5 are characterized, while for n≥9 the graphs G with δ=2 achieving the bound Γ_pr(G)=3n/4 are characterized.     

Distance graphs on R n with 1-norm

Suppose S is a subset of a metric space X with metric d. For each subset D⊆{d(x,y):x,y∈S,x≠y}, the distance graph G(S,D) is the graph with vertex set S and edge set E(S,D)={xy:x,y∈S,d(x,y)∈D}. The current paper studies distance graphs on the n-space R ₁ ⁿ with 1-norm. In particular, most attention is paid to the subset Z ₁ ⁿ of all lattice points of R ₁ ⁿ . The results obtained include the degrees of vertices, components, and chromatic numbers of these graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Supported in part by the National Science Council under grant NSC-94-2115-M-002-015. Taida Institue for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.