A degree condition for spanning eulerian subgraphs

Let p ≥ 2 be a fixed integer. Let G be a simple and 2-edge-connected graph on n vertices, and let g be the girth of G. If d(u) + d(v) ≥ (2/(g ? 2))((n/p) ? 4 + g) holds whenever uv ? E(G), and if n is sufficiently large compared to p, then either G has a spanning eulerian subgraph or G can be contracted to a graph G₁ of order at most p without a spanning eulerian subgraph. Furthermore, we characterize the graphs that satisfy the conditions above such that G₁ has order p and does not have any spanning eulerian subgraph. © 1993 John Wiley & Sons, Inc.     

Up-embeddability via girth and the degree-sum of adjacent vertices

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d _G (u) + d _G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d _G (u) + d _G (v) ⩾ n − 2g + 3 (resp. d _G (u) + d _G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight. This work was supported by National Natural Science Foundation of China (Grant No. 10571013)     

On the Wiener Index of Graphs

The Wiener index of a graph G is defined as W(G)=∑ _u,v d _G (u,v), where d _G (u,v) is the distance between u and v in G and the sum goes over all the pairs of vertices. In this paper, we first present the 6 graphs with the first to the sixth smallest Wiener index among all graphs with n vertices and k cut edges and containing a complete subgraph of order n−k; and then we construct a graph with its Wiener index no less than some integer among all graphs with n vertices and k cut edges.     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

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.     

On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

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.     

Degree-Light-Free Graphs and Hamiltonian Cycles

 Let G and H be graphs. G is said to be degree-light H-free if G is either H-free or, for every induced subgraph K of G with K≅H, and every {u,v}⊆K, d i s t _K (u,v)=2 implies max {d(u),d(v)}≥|V(G)|/2. In this paper, we will show that every 2-connected graph with either degree-light {K _1,3, P ₆}-free or degree-light {K _1,3, Z}-free is hamiltonian (with three exceptional graphs), where P ₆ is a path of order 6 and Z is obtained from P ₆ by adding an edge between the first and the third vertex of P ₆ (see Figure 1). Received: December 9, 1998?Final version received: July 21, 1999     

Counterexamples to a conjecture of mader about cycles through specified vertices inn-edge-connected graphs

For each oddn≥3, we constructn-edge-connected graphsG with the following property: There are two verticesu andv inG such that for every cycleC inG passign throughu andv the graphG-E(C) is not (n-2)-edge-connected. HereE(C) denotes the set of edges ofC, and a cycle is allowed to pass through a vertex more than once.     

A charcterization of 4-edge-connected eulerian graphs

Let v be an arbitrary vertex of a 4-edge-connected Eulerian graph G. First we show the existence of a nonseparating cycle decompositiion of G with respect to v. With the help of this decomposition we are then able to construct 4 edge-independent spanning trees with the common root v in the sam graph. We conclude that an Eulerian graph G is 4-edge-connected iff for every vertex r ? V (G) there exist 4 edge-independent spanning trees with a common root r. © 1996 John Wiley & Sons, Inc.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

A Note on L（2, 1）-labelling of Trees

An L(2,1)-labelling of a graph G is a function from the vertex set V (G) to the set of all nonnegative integers such that |f(u) f(v)| ≥ 2 if d G (u,v)=1 and |f(u) f(v)| ≥ 1 if d G (u,v)=2.The L(2,1)-labelling problem is to find the smallest number,denoted by λ(G),such that there exists an L(2,1)-labelling function with no label greater than it.In this paper,we study this problem for trees.Our results improve the result of Wang [The L(2,1)-labelling of trees,Discrete Appl.Math.154 (2006) 598-603].     

Extremal Problems for Imbalanced Edges

Albertson [2] has introduced the imbalance of an edge e=uv in a graph G as |d_G(u)−d_G(v)|. If for a graph G of order n and size m the minimum imbalance of an edge of G equals d, then our main result states that with equality if and only if G is isomorphic to We also prove best-possible upper bounds on the number of edges uv of a graph G such that |d_G(u)−d_G(v)|≥d for some given d.     

On 2-Detour Subgraphs of the Hypercube

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d _H (x, y) ≤ d _G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q _n has at least clog₂ n · 2 ⁿ edges. József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398. Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.     

New Local Conditions for a Graph to be Hamiltonian

For a vertex w of a graph G the ball of radius 2 centered at w is the subgraph of G induced by the set M₂(w) of all vertices whose distance from w does not exceed 2. We prove the following theorem: Let G be a connected graph where every ball of radius 2 is 2-connected and d(u)+d(v)≥|M₂(w)|−1 for every induced path uwv. Then either G is hamiltonian or for some p≥2 where ∨ denotes join. As a corollary we obtain the following local analogue of a theorem of Nash-Williams: A connected r-regular graph G is hamiltonian if every ball of radius 2 is 2-connected and for each vertex w of G. Supported by the Swedish Research Council (VR)     

Up-Embeddability of a Graph by Order and Girth

Let G be a connected graph of order n and girth g. If d_G(u) + d_G(v) ≥ n − 2g + 5 for any two non-adjacent vertices u and v, then G is up-embeddable. Further more, the lower bound is best possible. Similarly the result of k-edge connected simple graph with girth g is also obtained, k = 2,3. Partially supported by the Postdoctoral Seience Foundation of Central South University and NNSFC under Grant No. 10751013.     

Fan-Type Theorem for Long Cycles Containing a Specified Edge

In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u),d(v)}:d(u,v)=2}=μ, then for each edge e, G has a cycle containing e of length at least min{n,2μ}, unless G is a spanning subgraph of K_μ + K^c_n−μ or K₃+(lK_μ−2∪K_s), where n=l(μ−2)+s+3,1≤s≤μ−2. Partially supported by NNSFC(No. 60172005); Partially supported by NNSFC(No. 10431020);