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.     

Distance two edge labelings of lattices

Suppose G is a graph. Two edges e and e′ in G are said to be adjacent if they share a common end vertex, and distance two apart if they are nonadjacent but both are adjacent to a common edge. Let j and k be two positive integers. An L(j,k)-edge-labeling of a graph G is an assignment of nonnegative integers, called labels, to the edges of G such that the difference between labels of any two adjacent edges is at least j, and the difference between labels of any two edges that are distance two apart is at least k. The minimum range of labels over all L(j,k)-edge-labelings of a graph G is called the L(j,k)-edge-labeling number of G, denoted by $\lambda_{j,k}'(G)$ . Let m, j and k be positive integers. An m-circular-L(j,k)-edge-labeling of a graph G is an assignment f from {0,1,…,m?1} to the edges of G such that, for any two edges e and e′, |f(e)?f(e′)| _m ≥j if e and e′ are adjacent, and |f(e)?f(e′)| _m ≥k if e and e′ are distance two apart, where |a| _m =min{a,m?a}. The minimum m such that G has an m-circular-L(j,k)-edge-labeling is called the circular-L(j,k)-edge-labeling number of G, denoted by $\sigma_{j,k}'(G)$ . This paper investigates the L(1,1)-edge-labeling numbers, the L(2,1)-edge-labeling numbers and the circular-L(2,1)-edge-labeling numbers of the hexagonal lattice, the square lattice, the triangular lattice and the strong product of two infinite paths.     

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 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.     

L(2,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(2,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(2,1)-labelings of the edge-path-replacement G(P ₃) of a graph, called (2,1)-total labeling of G, was introduced by Havet and Yu in 2002 (Workshop graphs and algorithms, Dijon, France, 2003; Discrete Math. 308:498–513, 2008). They (Havet and Yu, Discrete Math. 308:498–513, 2008) obtain the bound $\Delta+1\leq\lambda^{T}_{2}(G)\leq2\Delta+1$ and conjectured $\lambda^{T}_{2}(G)\leq\Delta+3$ . In this paper, we obtain that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ.     

L(d,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(d,1)-labelings of the edge-path-replacement G(P _k ) of a graph, called (d,1)-total labeling of G, was introduced in 2002 by Havet and Yu (Workshop graphs and algorithms, 2003; Discrete Math 308:493–513, 2008). Havet and Yu (Discrete Math 308:498–513, 2008) obtained the bound $\Delta+ d-1\leq\lambda^{T}_{d}(G)\leq2\Delta+ d-1$ and conjectured $\lambda^{T}_{d}(G)\leq\Delta+2d-1$ . In (Lü in J Comb Optim, to appear; Zhejiang University, submitted), we worked on L(2,1)-labelings-number and L(1,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G, and obtained that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ. In this paper, we will study L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G for d≥3 and k≥4.     

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.     

Roman domination on strongly chordal graphs

Given real numbers b≥a>0, an (a,b)-Roman dominating function of a graph G=(V,E) is a function f:V→{0,a,b} such that every vertex v with f(v)=0 has a neighbor u with f(u)=b. An independent/connected/total (a,b)-Roman dominating function is an (a,b)-Roman dominating function f such that {v∈V:f(v)≠0} induces a subgraph without edges/that is connected/without isolated vertices. For a weight function $w{:} V\to\Bbb{R}$ , the weight of f is w(f)=∑ _v∈V w(v)f(v). The weighted (a,b)-Roman domination number $\gamma^{(a,b)}_{R}(G,w)$ is the minimum weight of an (a,b)-Roman dominating function of G. Similarly, we can define the weighted independent (a,b)-Roman domination number $\gamma^{(a,b)}_{Ri}(G,w)$ . In this paper, we first prove that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/independent (a,b)-Roman domination problems are NP-complete for bipartite graphs. We also show that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/weighted independent (a,b)-Roman domination problems are NP-complete for chordal graphs. We then give linear-time algorithms for the weighted (a,b)-Roman domination problem with b≥a>0, and the weighted independent (a,b)-Roman domination problem with 2a≥b≥a>0 on strongly chordal graphs with a strong elimination ordering provided.     

On backbone coloring of graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G,H) is a mapping f: V(G)→{1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(H) and |f(u)−f(v)|≥1 if uv∈E(G)\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring. In this paper, we characterize the backbone chromatic number of Halin graphs G=T∪C with respect to given spanning trees T. Also we study the backbone coloring for other special graphs such as complete graphs, wheels, graphs with small maximum average degree, graphs with maximum degree 3, etc.     

Fault-tolerant diameter for three family interconnection networks

Let G=(V,E) be an k-connected graph. The (k?1)-fault-tolerant diameter of G, denoted by D _k (G), is defined as D _k (G)=max?{d(G?F)|F?V(G)?and?|F|=k?1}, where G?F denotes the subgraph induced by V(G)?F. The fault-tolerant diameter is one important parameter for measuring the reliability and efficiency of interconnection networks. In this paper, we will give the bounds of fault-tolerant diameter of three family interconnection networks.     

On L(2,1)-labeling of generalized Petersen graphs

A variation of the classical channel assignment problem is to assign a radio channel which is a nonnegative integer to each radio transmitter so that ??close?? transmitters must receive different channels and ??very close?? transmitters must receive channels that are at least two channels apart. The goal is to minimize the span of a feasible assignment. This channel assignment problem can be modeled with distance-dependent graph labelings. A k-L(2,1)-labeling of a graph G is a mapping f from the vertex set of G to the set {0,1,2,??,k} such that |f(x)?f(y)|??2 if d(x,y)=1 and $f(x)\not =f(y)$ if d(x,y)=2, where d(x,y) is the distance between vertices x and y in G. The minimum k for which G admits an k-L(2,1)-labeling, denoted by ??(G), is called the ??-number of G. Very little is known about ??-numbers of 3-regular graphs. In this paper we focus on an important subclass of 3-regular graphs called generalized Petersen graphs. For an integer n??3, a graph G is called a generalized Petersen graph of order n if and only if G is a 3-regular graph consisting of two disjoint cycles (called inner and outer cycles) of length n, where each vertex of the outer (resp. inner) cycle is adjacent to exactly one vertex of the inner (resp. outer) cycle. In 2002, Georges and Mauro conjectured that ??(G)??7 for all generalized Petersen graphs G of order n??7. Later, Adams, Cass and Troxell proved that Georges and Mauro??s conjecture is true for orders 7 and 8. In this paper it is shown that Georges and Mauro??s conjecture is true for generalized Petersen graphs of orders 9, 10, 11 and 12.     

Hardness results and approximation algorithm for total liar’s domination in graphs

In this paper, we initiate the study of total liar’s domination of a graph. A subset L?V of a graph G=(V,E) is called a total liar’s dominating set of G if (i) for all v∈V, |N _G (v)∩L|≥2 and (ii) for every pair u,v∈V of distinct vertices, |(N _G (u)∪N _G (v))∩L|≥3. The total liar’s domination number of a graph G is the cardinality of a minimum total liar’s dominating set of G and is denoted by γ _TLR (G). The Minimum Total Liar’s Domination Problem is to find a total liar’s dominating set of minimum cardinality of the input graph G. Given a graph G and a positive integer k, the Total Liar’s Domination Decision Problem is to check whether G has a total liar’s dominating set of cardinality at most k. In this paper, we give a necessary and sufficient condition for the existence of a total liar’s dominating set in a graph. We show that the Total Liar’s Domination Decision Problem is NP-complete for general graphs and is NP-complete even for split graphs and hence for chordal graphs. We also propose a 2(lnΔ(G)+1)-approximation algorithm for the Minimum Total Liar’s Domination Problem, where Δ(G) is the maximum degree of the input graph G. We show that Minimum Total Liar’s Domination Problem cannot be approximated within a factor of $(\frac{1}{8}-\epsilon)\ln(|V|)$ for any ?>0, unless NP?DTIME(|V|^loglog|V|). Finally, we show that Minimum Total Liar’s Domination Problem is APX-complete for graphs with bounded degree 4.     

A note on (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-relaxed <Emphasis Type="Italic">L</Emphasis>(2, 1)-labeling of graphs

Let \(G=(V, E)\) be a graph. For two vertices u and v in G, we denote \(d_G(u, v)\) the distance between u and v. A vertex v is called an i-neighbor of u if \(d_G(u,v)=i\). Let s, t and k be nonnegative integers. An (s, t)-relaxed k-L(2, 1)-labeling of a graph G is an assignment of labels from \(\{0, 1, \ldots , k\}\) to the vertices of G if the following three conditions are met: (1) adjacent vertices get different labels; (2) for any vertex u of G, there are at most s 1-neighbors of u receiving labels from \(\{f(u)-1,f(u)+1\}\); (3) for any vertex u of G, the number of 2-neighbors of u assigned the label f(u) is at most t. The (s, t)-relaxed L(2, 1)-labeling number \(\lambda _{2,1}^{s,t}(G)\) of G is the minimum k such that G admits an (s, t)-relaxed k-L(2, 1)-labeling. In this article, we refute Conjecture 4 and Conjecture 5 stated in (Lin in J Comb Optim. doi: 10.1007/s10878-014-9746-9, 2013).     

On the mod sum number of H m,n

Let N denote the set of all positive integers. The sum graph G ⁺(S) of a finite subset S?N is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an mod sum graph if it is isomorphic to the sum graph of some S?Z _M \{0} and all arithmetic performed modulo M where M≥|S|+1. The mod sum number ρ(G) of G is the smallest number of isolated vertices which when added to G result in a mod sum graph. It is known that the graphs H _m,n (n>m≥3) are not mod sum graphs. In this paper we show that H _m,n are not mod sum graphs for m≥3 and n≥3. Additionally, we prove that ρ(H _m,3)=m for m≥3, H _m,n ∪ρK ₁ is exclusive for m≥3 and n≥4 and $m(n-1) \leq \rho(H_{m,n})\leq \frac{1}{2} mn(n-1)$ for m≥3 and n≥4.     

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.     

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.     

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).     

Star list chromatic number of planar subcubic graphs

A proper coloring of the vertices of a graph G is called a star-coloring if the union of every two color classes induces a star forest. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring π such that π(v)∈L(v). If G is L-star-colorable for any list assignment L with |L(v)|≥k for all v∈V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by $\chi_{s}^{l}(G)$ , is the smallest integer k such that G is k-star-choosable. In this paper, we prove that every planar subcubic graph is 6-star-choosable.     

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.     

Note on the hardness of generalized connectivity

Let G be a nontrivial connected graph of order n and let k be an integer with 2??k??n. For a set S of k vertices of G, let ??(S) denote the maximum number ? of edge-disjoint trees T ₁,T ₂,??,T _? in G such that V(T _i )??V(T _j )=S for every pair i,j of distinct integers with 1??i,j???. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by ?? _k (G), of G is defined by ?? _k (G)=min{??(S)}, where the minimum is taken over all k-subsets S of V(G). Thus ?? ₂(G)=??(G), where ??(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k ₁ and k ₂, given a graph G and a k ₁-subset S of V(G), the problem of deciding whether G contains k ₂ internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k ₁ is a fixed integer of at least 4, but k ₂ is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k ₂ is a fixed integer of at least 2, but k ₁ is not a fixed integer, we show that the problem also becomes NP-complete.