Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d^w(u)+d^w(v)2d for every pair of non-adjacent vertices u and vV(G) x,y . Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.     

On -connected graphs

A graph is called -connected if is -edge-connected and is -edge-connected for every vertex . The study of -connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a -vertex-connected orientation if and only if it is (2,2)-connected. In this paper, we provide a construction of the family of -connected graphs for even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)-connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph and an integer , what is the minimum number of edges to be added to make -connected. Both these results are based on a new splitting-off theorem for -connected graphs.     

On the weights of simple paths in weighted complete graphs

Consider a weighted simple graph G on the vertex set {1,..., n}. For any path p in G, we call ω _G (p) the sum of the weights of the edges of the path, and, for any {i, j} ? {1,..., n}, we define the multiset $$D_{\{ i,j\} } (G) = \left\{ {\left. {w_G \left( p \right)} \right|p a simple path between i and j} \right\}.$$ We establish a criterion for a multisubset of ? to be of the form $\mathcal{D}_{\{ i,j\} } (G)$ for some weighted complete graph G and for some i, j vertices of G. Besides we establish a criterion for a family of multi-subsets of ? to be of the form for some weighted complete graph G with vertex set {1,..., n}.     

On extremal weighted digraphs with no heavy paths

Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs.     

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G - V(S).For an integer k ≥ 4,...     

赋权图中的路和圈

本文研究了赋权图中的最长路和最长圈,将关于非赋权图中最长路和最长圈的一些结果推广到赋权图上.     

Cycles in 2-connected graphs

Let be a class of graphs on n vertices. For an integer c, let be the smallest integer such that if G is a graph in with more than edges, then G contains a cycle of length more than c. A classical result of Erdös and Gallai is that if is the class of all simple graphs on n vertices, then . The result is best possible when n-1 is divisible by c-1, in view of the graph consisting of copies of K_c all having exactly one vertex in common. Woodall improved the result by giving best possible bounds for the remaining cases when n-1 is not divisible by c-1, and conjectured that if is the class of all 2-connected simple graphs on n vertices, thenwhere , 2tc/2, is the number of edges in the graph obtained from K_c+1-t by adding n-(c+1-t) isolated vertices each joined to the same t vertices of K_c+1-t. By using a result of Woodall together with an edge-switching technique, we confirm Woodall's conjecture in this paper.     

2-connected coverings of bounded degree in 3-connected graphs

In a recent paper, Barnette showed that every 3-connected planar graph has a 2-connected spanning subgraph of maximum degree at most fifteen, he also constructed a planar triangulation that does not have 2-connected spanning subgraphs of maximum degree five. In this paper, we show that every 3-connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2-connected spanning subgraph of maximum degree at most six. © 1995 John Wiley & Sons, Inc.     

On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.     

On weighted lattice paths

A weighted lattice path from (1, 1) to (n, m) is a path consisting of unit vertical, horizontal, and diagonal steps of weight w. Let f(0), f(1), f(2), … be a nondecreasing sequence of positive integers; the path connecting the points of the set $\text{{(n, m) ¦ f(n ? 1) ? m ? f(n), n = 1, 2, …}}$ will be called the roof determined by f. We determine the number of weighted lattice paths from (1, 1) to (n + 1, f(n)) which do not cross the roof determined by f. We also determine the polynomials that must be placed in each cell below the roof such that if a 1 is placed in each cell whose lower left-hand corner is a point of the roof, every k × k square subarray comprised of adjacent rows and columns and containing at least one 1 will have determinant $\text{x(}\text{k}\text{2}\text{)}$.     

On paths in planar graphs

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y and containing any specified edge other than xy. © 1997 John Wiley & Sons, Inc.     

4-factors in 2-connected star-free graphs

Let t≥3 be an integer. We show that if G is a 2-connected K_1,t-free graph with minimum degree at least (3t+1)/2, then G has a 4-factor.     

On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ₁ components and κ₂ blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|−|E(H)|(κ₁−1)+β(κ₂−1) for all (,β) such that +β5,2+5β20, and β3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails.     

Hamiltonicity of regular 2-connected graphs

Let G be a k-regular 2-connected graph of order n. Jackson proved that G is hamiltonian if n ≤ 3k. Zhu and Li showed that the upper bound 3k on n can be relaxed to 22/7k if G is 3-connected and k ≥ 63. We improve both results by showing that G is hamiltonian if n ≤ 7/2k − 7 and G does not belong to a restricted class F of nonhamiltonian graphs of connectivity 2. To establish this result we obtain a variation of Woodall's Hopping Lemma and use it to prove that if n ≤ 7/2k − 7 and G has a dominating cycle (i.e., a cycle such that the vertices off the cycle constitute an independent set), then G is hamiltonian. We also prove that if n ≤ 4k − 3 and G ∉ F, then G has a dominating cycle. For k ≥ 4 it is conjectured that G is hamiltonian if n ≤ 4k and G ∉ F. © 1996 John Wiley & Sons, Inc.