An Implicit Degree Condition for Pancyclicity of Graphs

Zhu, Li and Deng introduced in 1989 the definition of implicit degree of a vertex v in a graph G, denoted by id(v), by using the degrees of the vertices in its neighborhood and the second neighborhood. And they obtained sufficient conditions with implicit degrees for a graph to be hamiltonian. In this paper, we prove that if G is a 2–connected graph of order n ≥ 3 such that id(v) ≥ n/2 for each vertex v of G, then G is pancyclic unless G is bipartite, or else n = 4r, r ≥ 2 and G is in a class of graphs F _4r defined in the paper.     

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.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.     

On upper domination Ramsey numbers for graphs

The classical Ramsey number r(m,n) can be defined as the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, β(B)⩾m or β(R)⩾n, where β(G) denotes the independence number of a graph G. We define the upper domination Ramsey number u(m,n) as the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, Γ(B)⩾m or Γ(R)⩾n, where Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. The mixed domination Ramsey number v(m,n) is defined to be the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, Γ(B)⩾m or β(R)⩾n. Since β(G)⩽Γ(G) for every graph G, u(m,n)⩽v(m,n)⩽r(m,n). We develop techniques to obtain upper bounds for upper domination Ramsey numbers of the form u(3,n) and mixed domination Ramsey numbers of the form v(3,n). We show that u(3,3)=v(3,3)=6, u(3,4)=8, v(3,4)=9, u(3,5)=v(3,5)=12 and u(3,6)=v(3,6)=15.     

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex v such that d(v, v ) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u, v) + d(u, v ) = diam G for all u, v ∈ V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d – 1 and conjectured that n ≥ 4d – 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d – 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented.     

Two edge-disjoint hamiltonian cycles in graphs

E. Schmeichel and D. Hayes showed that ifG is a 2-connected graph withd(u) +d(v)≥n ?1 for every pair of nonadjacent vertices andv, then G has a Hamiltonian cycle unlessG is the graph of Fig. 2 (b). In this paper, it is proved that, under almost the same conditions as Schmeichel and Hayes’s Theorem, namely,G is a 2-connected graph of ordern (n ≥ 40) with δ(G) ≥ 7 for every pair of nonadjacent vertices andv, G has two edge-disjoint Hamiltonian cycles unlessG is one of the graphs in Fig. 1 or Fig. 2, and this conclusion is best possible.     

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     

An implicit degree condition for relative length of long paths and cycles in graphs

For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) ? 1.     

Convexity in oriented graphs

For vertices u and v in an oriented graph D, the closed interval I[u,v] consists of u and v together with all vertices lying in a u−v geodesic or v−u geodesic in D. For S⊆V(D), I[S] is the union of all closed intervals I[u,v] with u,v∈S. A set S is convex if I[S]=S. The convexity number con(D) is the maximum cardinality of a proper convex set of V(D). The nontrivial connected oriented graphs of order n with convexity number n−1 are characterized. It is shown that there is no connected oriented graph of order at least 4 with convexity number 2 and that every pair k, n of integers with 1⩽k⩽n−1 and k≠2 is realizable as the convexity number and order, respectively, of some connected oriented graph. For a nontrivial connected graph G, the lower orientable convexity number con⁻(G) is the minimum convexity number among all orientations of G and the upper orientable convexity number con⁺(G) is the maximum such convexity number. It is shown that con⁺(G)=n−1 for every graph G of order n⩾2. The lower orientable convexity numbers of some well-known graphs are determined, with special attention given to outerplanar graphs.     

On the Szeged and the Laplacian Szeged spectrum of a graph

For a given graph G its Szeged weighting is defined by w(e)=n_u(e)n_v(e), where e=uv is an edge of G,n_u(e) is the number of vertices of G closer to u than to v, and n_v(e) is defined analogously. The adjacency matrix of a graph weighted in this way is called its Szeged matrix. In this paper we determine the spectra of Szeged matrices and their Laplacians for several families of graphs. We also present sharp upper and lower bounds on the eigenvalues of Szeged matrices of graphs.     

The Chartrand-Schuster conjecture: Graphs with unique distance trees are regular

Let G be a finite connected graph with no cut vertex. A distance tree T is a spanning tree of G which further satisfies the condition that for some vertex v, d_G(v, u) = d_T(v, u) for all u, where d_G(v, u) denotes the distance of u from v in the graph G. The conjecture that if all distance trees of G are isomorphic to each other then G is a regular graph, is settled affirmatively. The conjecture was made by Chartrand and Schuster.     

A short proof of Fan's theorem

In this note, we give a new short proof of the following theorem: Let G be a 2-connected graph of order n. If for any two vertices u and v with d(u,v)=2,max{d(u),d(v)}?c/2, then the circumference of G is at least c, where 3?c?n and d(u,v) is the distance between u and v in G.     

Minimal rankings and the arank number of a path

Given a graph G, a function f:V(G)→{1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)>f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψ_r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψ_r(P_n), a problem posed by Laskar and Pillone in 2000.     

Constructing and classifying neighborhood anti-Sperner graphs

For a simple graph G let N_G(u) be the (open) neighborhood of vertex u∈V(G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v∈V(G)?{u} with N_G(u)⊆N_G(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if N_H(u)≠N_H(v) when u≠v, for all u and v∈V(H).In Porter and Yucas [T.D. Porter, J.L. Yucas. Graphs whose vertex-neighborhoods are anti-sperner, Bulletin of the Institute of Combinatorics and its Applications 44 (2005) 69-77] a characterization of regular NAS graphs was given: ‘each regular NAS graph can be obtained from a host graph by replacing vertices by null graphs of appropriate sizes, and then joining these null graphs in a prescribed manner’. We extend this characterization to all NAS graphs, and give algorithms to construct all NAS graphs from host ND graphs. Then we find and classify all connected r-regular NAS graphs for r=0,1,…,6.     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

An efficient condition for a graph to be Hamiltonian

Let G=(V,E) be a 2-connected simple graph and let d_G(u,v) denote the distance between two vertices u,v in G. In this paper, it is proved: if the inequality d_G(u)+d_G(v)?|V(G)|-1 holds for each pair of vertices u and v with d_G(u,v)=2, then G is Hamiltonian, unless G belongs to an exceptional class of graphs. The latter class is described in this paper. Our result implies the theorem of Ore [Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55]. However, it is not included in the theorem of Fan [New sufficient conditions for cycles in graph, J. Combin. Theory Ser. B 37 (1984) 221-227].     

Trees with second minimum general Randić index for α>0

The general Randi? index R _α (G) of a graph G is the sum of the weights (d(u)d(v)) ^α of all edges uv of G, where α is a real number(α≠0) and d(u) denotes the degree of the vertex u. We have known that P _n has minimum general Randi? index for α>0 among trees when n≥5. In this paper, we prove that P _n,3 has second minimum general Randi? index for α>0 among trees when n≥7.