Minimum implicit degree condition restricted to claws for hamiltonian cycles

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian.     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

An implicit degree ore-condition for pancyclicity of graphs

In 1989, Zhu, Li and Deng introduced the definition of implicit degree of a vertex v in a graph G, denoted by id(v). In this paper, we prove that if G is a 2-connected graph of order n such that id(u) + id(v) ≥ n for each pair of nonadjacent vertices u and v in G, then G is pancyclic unless G is bipartite, or else n = 4r, r ≥ 2 and G is isomorphic to F4r .     

Long Paths Through Specified Vertices In 3-Connected Graphs

Let h ≥ 6 be an integer, let G be a 3-connected graph with ∣V(G)∣ ≥ h − 1, and let x and z be distinct vertices of G. We show that if for any nonadjacent distinct vertices u and v in V(G) − {x, z}, the sum of the degrees of u and v in G is greater than or equal to h, then for any subset Y of V(G) − {x, z} with ∣Y∣ ≤ 2, G contains a path which has x and z as its endvertices, passes through all vertices in Y, and has length at least h − 2. We also show a similar result for cycles in 2-connected graphs.     

Characterization of digraphs with equal domination graphs and underlying graphs

A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u,v}∈E[dom(D)] whenever (u,z)∈A(D) or (v,z)∈A(D) for every other vertex z∈V(D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)=dom(D). The maximum and minimum number of single arcs in these digraphs, and their characteristics, is given.     

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

On circuits and pancyclic line graphs

Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n – 1 – p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n – 1 – p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G) ≥ 2 and deg u + deg v ≥ n – 1 for every pair of nonadjacent vertices u and v. The bound n – 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.     

Survey of Double Vertex Graphs

 Let G be a (V,E) graph of order p≥2. The double vertex graph U ₂ (G) is the graph whose vertex set consists of all 2-subsets of V such that two distinct vertices {x,y} and {u,v} are adjacent if and only if |{x,y}∩{u,v}|=1 and if x=u, then y and v are adjacent in G. For this class of graphs we discuss the regularity, eulerian, hamiltonian, and bipartite properties of these graphs. A generalization of this concept is n-tuple vertex graphs, defined in a manner similar to double vertex graphs. We also review several recent results for n-tuple vertex graphs. Received: October, 2001 Final version received: September 20, 2002 Dedicated to Frank Harary on the occasion of his Eightieth Birthday and the Manila International Conference held in his honor     

A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000     

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Pancyclic oriented graphs

Let D be an oriented graph of order n ≧ 9 and minimum degree n ? 2. This paper proves that D is pancyclic if for any two vertices u and v, either uv ? A(D), or d_D⁺(u) + d_D^?(v) ≧ n ? 3.     

Two Conjectures on Graceful Digraphs

A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from {0,1, … , q} to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) = (g(v)? g(u))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. Bloom and Hsu (SIAM J Alg Discr Methods 6:519–536, 1985) conjectured that, all unicyclic wheels are graceful. Also, Zhao et al. (J Prime Res Math 4:118–126, 2008) conjectured that, for any positive even n and any integer m ≥ 14, the digraph ${n-\overrightarrow{C_m}}$ is graceful. In this paper, we prove both the conjectures.     

THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS

51.IntroductionandNotationsLetD=(V,E)beadigraphandL(D)denotethesetofcyclelengthsofD.ForuEVandintegeri21,letfo(u):={vEVIthereedestsadirectedwalkoflengthifromutov}.WedelveRo(u):={u}.Letu,vEV.IfN (v)=N (v)andN--(v)=N--(v),thenwecanvacopyofu.LotDbeaprimitivedigraphand7(D)denotetheexponentofD.In1950,H.WielandtI61foundthat7(D)5(n--1)' 1andshowedthatthereisapiquedigraphthatattainsthisbound.In1964,A.L.DulmageandN.S.Mendelsohn[2]ObservedthattherearegapsintheexponentsetEd={ry(D)IDEPD.}…     

The eavesdropping number of a graph

Let G be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices u and v, a minimum {u, v}-separating set is a smallest set of edges in G whose removal disconnects u and v. The edge connectivity of G, denoted λ(G), is defined to be the minimum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. We introduce and investigate the eavesdropping number, denoted ε(G), which is defined to be the maximum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. Results are presented for regular graphs and maximally locally connected graphs, as well as for a number of common families of graphs.     

On the Super Edge-Magic Deficiency of Graphs

A (p, q) graph G is edge-magic if there exists a bijective function f: V(G) ∪ E(G) → {1,2,…,p + q} such that f(u) + f(v) + f(uv) = k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V(G)) = {1,2,…,p}. The question studied in this paper is for which graphs is it possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic? If it is possible for a given graph G, then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, μ_s(G) of G; otherwise we define it to be + ∞.     

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2‐colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α‐size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α‐size of trees with maximum degree three. Let α₃(n) be the smallest α‐size among all trees with n vertices, each of degree at most three. We prove that α₃(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α₃(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999     

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.