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.     

Notes on L(1,1) and L(2,1) labelings for n-cube

Suppose \(d\) is a positive integer. An \(L(d,1)\) -labeling of a simple graph \(G=(V,E)\) is a function \(f:V\rightarrow \mathbb{N }=\{0,1,2,{\ldots }\}\) such that \(|f(u)-f(v)|\ge d\) if \(d_G(u,v)=1\) ; and \(|f(u)-f(v)|\ge 1\) if \(d_G(u,v)=2\) . The span of an \(L(d,1)\) -labeling \(f\) is the absolute difference between the maximum and minimum labels. The \(L(d,1)\) -labeling number, \(\lambda _d(G)\) , is the minimum of span over all \(L(d,1)\) -labelings of \(G\) . Whittlesey et al. proved that \(\lambda _2(Q_n)\le 2^k+2^{k-q+1}-2,\) where \(n\le 2^k-q\) and \(1\le q\le k+1\) . As a consequence, \(\lambda _2(Q_n)\le 2n\) for \(n\ge 3\) . In particular, \(\lambda _2(Q_{2^k-k-1})\le 2^k-1\) . In this paper, we provide an elementary proof of this bound. Also, we study the \(L(1,1)\) -labeling number of \(Q_n\) . A lower bound on \(\lambda _1(Q_n)\) are provided and \(\lambda _1(Q_{2^k-1})\) are determined.     

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.     

On (s,t)-relaxed L(2,1)-labeling of graphs

We initiate the study of relaxed \(L(2,1)\)-labelings of graphs. Suppose \(G\) is a graph. Let \(u\) be a vertex of \(G\). A vertex \(v\) is called an \(i\)-neighbor of \(u\) if \(d_G(u,v)=i\). A \(1\)-neighbor of \(u\) is simply called a neighbor of \(u\). Let \(s\) and \(t\) be two nonnegative integers. Suppose \(f\) is an assignment of nonnegative integers to the vertices of \(G\). If the following three conditions are satisfied, then \(f\) is called an \((s,t)\)-relaxed \(L(2,1)\)-labeling of \(G\): (1) for any two adjacent vertices \(u\) and \(v\) of \(G, f(u)\not =f(v)\); (2) for any vertex \(u\) of \(G\), there are at most \(s\) 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 minimum span of \((s,t)\)-relaxed \(L(2,1)\)-labelings of \(G\) is called the \((s,t)\)-relaxed \(L(2,1)\)-labeling number of \(G\), denoted by \(\lambda ^{s,t}_{2,1}(G)\). It is clear that \(\lambda ^{0,0}_{2,1}(G)\) is the so called \(L(2,1)\)-labeling number of \(G\). \(\lambda ^{1,0}_{2,1}(G)\) is simply written as \(\widetilde{\lambda }(G)\). This paper discusses basic properties of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of graphs. For any two nonnegative integers \(s\) and \(t\), the exact values of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of paths, cycles and complete graphs are determined. Tight upper and lower bounds for \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of complete multipartite graphs and trees are given. The upper bounds for \((s,1)\)-relaxed \(L(2,1)\)-labeling number of general graphs are also investigated. We introduce a new graph parameter called the breaking path covering number of a graph. A breaking path \(P\) is a vertex sequence \(v_1,v_2,\ldots ,v_k\) in which each \(v_i\) is adjacent to at least one vertex of \(v_{i-1}\) and \(v_{i+1}\) for \(i=2,3,\ldots ,k-1\). A breaking path covering of \(G\) is a set of disjoint such vertex sequences that cover all vertices of \(G\). The breaking path covering number of \(G\), denoted by \(bpc(G)\), is the minimum number of breaking paths in a breaking path covering of \(G\). In this paper, it is proved that \(\widetilde{\lambda }(G)= n+bpc(G^{c})-2\) if \(bpc(G^{c})\ge 2\) and \(\widetilde{\lambda }(G)\le n-1\) if and only if \(bpc(G^{c})=1\). The breaking path covering number of a graph is proved to be computable in polynomial time. Thus, if a graph \(G\) is of diameter two, then \(\widetilde{\lambda }(G)\) can be determined in polynomial time. Several conjectures and problems on relaxed \(L(2,1)\)-labelings are also proposed.     

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.     

The (vertex-)monochromatic index of a graph

A tree T in an edge-colored (vertex-colored) graph H is called a monochromatic (vertex-monochromatic) tree if all the edges (internal vertices) of T have the same color. For \(S\subseteq V(H)\), a monochromatic (vertex-monochromatic) S-tree in H is a monochromatic (vertex-monochromatic) tree of H containing the vertices of S. For a connected graph G and a given integer k with \(2\le k\le |V(G)|\), the k -monochromatic index \(mx_k(G)\) (k -monochromatic vertex-index \(mvx_k(G)\)) of G is the maximum number of colors needed such that for each subset \(S\subseteq V(G)\) of k vertices, there exists a monochromatic (vertex-monochromatic) S-tree. For \(k=2\), Caro and Yuster showed that \(mc(G)=mx_2(G)=|E(G)|-|V(G)|+2\) for many graphs, but it is not true in general. In this paper, we show that for \(k\ge 3\), \(mx_k(G)=|E(G)|-|V(G)|+2\) holds for any connected graph G, completely determining the value. However, for the vertex-version \(mvx_k(G)\) things will change tremendously. We show that for a given connected graph G, and a positive integer L with \(L\le |V(G)|\), to decide whether \(mvx_k(G)\ge L\) is NP-complete for each integer k such that \(2\le k\le |V(G)|\). Finally, we obtain some Nordhaus–Gaddum-type results for the k-monochromatic vertex-index.     

On the efficiency index of a graph

A graph \(G\) has an efficient dominating set \(D \subseteq V(G)\) if \(D\) dominates every vertex exactly once. In this paper we introduce the study of the family \({S_k}\) of graphs for which every \(G-S\) is efficiently dominatable for \(0 \le |S|\le k\). Assuming that \(G\) is efficiently dominatable, the efficiency index is the largest value k for which \(G\) is in \(S_k\). A graph \(G\) will be called super-efficient if every induced subgraph is efficiently dominatable. We give some characterizations for trees, grids, cylinders and torii to be super-efficient.     

On the most imbalanced orientation of a graph

We study the problem of orienting the edges of a graph such that the minimum over all the vertices of the absolute difference between the outdegree and the indegree of a vertex is maximized. We call this minimum the imbalance of the orientation, i.e. the higher it gets, the more imbalanced the orientation is. The studied problem is denoted by \({{\mathrm{\textsc {MaxIm}}}}\). We first characterize graphs for which the optimal objective value of \({{\mathrm{\textsc {MaxIm}}}}\) is zero. Next we show that \({{\mathrm{\textsc {MaxIm}}}}\) is generally NP-hard and cannot be approximated within a ratio of \(\frac{1}{2}+\varepsilon \) for any constant \(\varepsilon >0\) in polynomial time unless \(\texttt {P}=\texttt {NP}\) even if the minimum degree of the graph \(\delta \) equals 2. Then we describe a polynomial-time approximation algorithm whose ratio is almost equal to \(\frac{1}{2}\). An exact polynomial-time algorithm is also derived for cacti. Finally, two mixed integer linear programming formulations are presented. Several valid inequalities are exhibited with the related separation algorithms. The performance of the strengthened formulations is assessed through several numerical experiments.     

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.     

Spanning tree of a multiple graph

Journal of Combinatorial Optimization - We study undirected multiple graphs of any natural multiplicity $$k>1$$ . There are edges of three types: ordinary edges, multiple edges and...     

The optimum traversal of a graph

For a given graph (network) having costs [c_ij] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called thegraph traversal problem, or theChinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem.This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links.The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity q_ij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results.There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.