An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Labeling trees with a condition at distance two

An L(h,k)-labeling of a graph G is an integer labeling of vertices of G, such that adjacent vertices have labels which differ by at least h, and vertices at distance two have labels which differ by at least k. The span of an L(h,k)-labeling is the difference between the largest and the smallest label. We investigate L(h,k)-labelings of trees of maximum degree Δ, seeking those with small span. Given Δ, h and k, span λ is optimal for the class of trees of maximum degree Δ, if λ is the smallest integer such that every tree of maximum degree Δ has an L(h,k)-labeling with span at most λ. For all parameters Δ,h,k, such that h<k, we construct L(h,k)-labelings with optimal span. We also establish optimal span of L(h,k)-labelings for stars of arbitrary degree and all values of h and k.     

Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E) be a finite, simple and non-empty (p,q)-graph of order p and size q. An (a,d)-vertex-antimagic total labeling is a bijection f from V(G)∪E(G) onto the set of consecutive integers 1,2,…,p+q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f(xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f(x). Such a labeling is called super if the smallest possible labels appear on the vertices.In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Distance-two labelings of digraphs

For positive integers j?k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|?j if x is adjacent to y in D and |f(x)-f(y)|?k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the -number of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies -numbers of digraphs. In particular, we determine -numbers of digraphs whose longest dipath is of length at most 2, and -numbers of ditrees having dipaths of length 4. We also give bounds for -numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining -numbers of ditrees whose longest dipath is of length 3.     

On the hole index of L(2,1)-labelings of r-regular graphs

An L(2,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G so that adjacent vertices get labels at least distance two apart and vertices at distance two get distinct labels. A hole is an unused integer within the range of integers used by the labeling. The lambda number of a graph G, denoted λ(G), is the minimum span taken over all L(2,1)-labelings of G. The hole index of a graph G, denoted ρ(G), is the minimum number of holes taken over all L(2,1)-labelings with span exactly λ(G). Georges and Mauro [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] conjectured that if G is an r-regular graph and ρ(G)?1, then ρ(G) must divide r. We show that this conjecture does not hold by providing an infinite number of r-regular graphs G such that ρ(G) and r are relatively prime integers.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

Finding monotone paths in edge-ordered graphs

An edge-ordering of a graph G=(V,E) is a one-to-one function f from E to a subset of the set of positive integers. A path P in G is called an f-ascent if f increases along the edge sequence of P. The heighth(f) of f is the maximum length of an f-ascent in G.In this paper we deal with computational problems concerning finding ascents in graphs. We prove that for a given edge-ordering f of a graph G the problem of determining the value of h(f) is NP-hard. In particular, the problem of deciding whether there is an f-ascent containing all the vertices of G is NP-complete. We also study several variants of this problem, discuss randomized and deterministic approaches and provide an algorithm for the finding of ascents of order at least k in graphs of order n in running time O(4^kn^O(1)).     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

Distance-two labellings of Hamming graphs

Let j≥k≥0 be integers. An ?-L(j,k)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…,?} such that |?(u)−?(v)|≥j if u,v are adjacent and |?(u)−?(v)|≥k if they are distance two apart. Let λ_j,k(G) be the smallest integer ? such that G admits an ?-L(j,k)-labelling. Define to be the smallest ? if G admits an ?-L(j,k)-labelling with ?(V)={0,1,2,…,?} and ∞ otherwise. An ?-cyclic L(j,k)-labelling is a mapping ?:V→Z_? such that |?(u)−?(v)|_?≥j if u,v are adjacent and |?(u)−?(v)|_?≥k if they are distance two apart, where |x|_?=min{x,?−x} for x between 0 and ?. Let σ_j,k(G) be the smallest ?−1 of such a labelling, and define similarly to . We determine λ_2,0, , σ_2,0 and for all Hamming graphs K_q1□K_q2□?□K_qd (d≥2, q₁≥q₂≥?≥q_d≥2) and give optimal labellings, with the only exception being for q≥4. We also prove the following “sandwich theorem”: If q₁ is sufficiently large then for any graph G between K_q1□K_q2 and K_q1□K_q2□?□K_qd, and moreover we give a labelling which is optimal for these eight invariants simultaneously.     

The profile of the Cartesian product of graphs

Given a graph G, a proper labelingf of G is a one-to-one function from V(G) onto {1,2,…,|V(G)|}. For a proper labeling f of G, the profile widthw_f(v) of a vertex v is the minimum value of f(v)−f(x), where x belongs to the closed neighborhood of v. The profile of a proper labelingfofG, denoted by P_f(G), is the sum of all the w_f(v), where v∈V(G). The profile ofG is the minimum value of P_f(G), where f runs over all proper labeling of G. In this paper, we show that if the vertices of a graph G can be ordered to satisfy a special neighborhood property, then so can the graph G×Q_n. This can be used to determine the profile of Q_n and K_m×Q_n.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

Independence polynomials of k-tree related graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and f_s(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f_s(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.     

On the minimum real roots of the adjoint polynomial of a graph

For a graph G, we denote by h(G,x) the adjoint polynomial of G. Let β(G) denote the minimum real root of h(G,x). In this paper, we characterize all the connected graphs G with .     

Pair lengths of product graphs

The pair length of a graph G is the maximum positive integer k, such that the vertex set of G can be partitioned into disjoint pairs {x,x^′}, such that d(x,x^′)?k for every x∈V(G) and x^′y^′ is an edge of G whenever xy is an edge. Chen asked whether the pair length of the cartesian product of two graphs is equal to the sum of their pair lengths. Our aim in this short note is to prove this result.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

The induced path function, monotonicity and betweenness

The geodesic interval function I of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebeský [20]. Surprisingly, Nebeský [23] showed that, if no further restrictions are imposed, the induced path function J of a connected graph G does not allow such an axiomatic characterization. Here J(u,v) consists of the set of vertices lying on the induced paths between u and v. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J. The function J satisfies betweenness if w∈J(u,v), with w≠u, implies u∉J(w,v) and x∈J(u,v) implies J(u,x)⊆J(u,v). It is monotone if x,y∈J(u,v) implies J(x,y)⊆J(u,v). In the case where we restrict ourselves to functions J that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .