Neighborhood Unions for the Existence of Disjoint Chorded Cycles in Graphs

A chorded cycle is a cycle with at least one chord. We prove that if G is a simple graph with order n ≥ 4k and ${|N_G(x)\cup N_G(y)|\geq 4k+1}$ for each nonadjacent pair of vertices x and y, then G contains k vertex-disjoint chorded cycles. The degree condition is sharp in general.     

On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs

Let r and s be nonnegative integers, and let G be a graph of order at least 3r + 4s. In Bialostocki et al. (Discrete Math 308:5886–5890, 2008), conjectured that if the minimum degree of G is at least 2r + 3s, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles, and they showed that the conjecture is true for r = 0, s = 2 and for s = 1. In this paper, we settle this conjecture completely by proving the following stronger statement; if the minimum degree sum of two nonadjacent vertices is at least 4r + 6s−1, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles.     

Large Vertex-Disjoint Cycles in a Bipartite Graph

Let s≥2 and k be two positive integers. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n≥s k and the minimum degree at least (s−1)k+1. When s=2 and n >2k, it is proved in [5] that G contains k vertex-disjoint cycles. In this paper, we show that if s≥3, then G contains k vertex-disjoint cycles of length at least 2s. Received: March 2, 1998 Revised: October 26, 1998     

路或圈的笛卡尔乘积图的支撑树数

设G是路或圈的笛卡尔乘积图，t(G)表示G的支撑树数.该文借助于第二类Chebyshev多项式给出t(G)的公式，并考虑了t(G)的线性递归关系及渐近性态.     

Zonotopal Subdivisions of Cyclic Zonotopes

The cyclic zonotope (n, d) is the zonotope in ^d generated by any n distinct vectors of the form (1, t, t ²,..., t ^d–1). It is proved that the refinement poset of all proper zonotopal subdivisions of (n, d) which are induced by the canonical projection : (n, d) (n, d), in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of (n, d) is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of (n, d) are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.     

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let denote the number of convex cycles of a simple graph G of order n, size m, and girth . It is proved that and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.     

k-Factors and Spanning Subgraph in Graphs

In this paper, we discussed k-factors and spanning subgraph, and propose a conjecture which will lead to a series of important conclusion.     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.     

生成树与图的结构

在网络研究中,人们需要将图分解为指定的结构,来研究网络的普适性、鲁棒又脆弱性,探索网络进化、动力学复杂性、节点多样性、时空演化复杂性等.生成树与图的结构得到研究,海林图,唯一圈图,具有特殊完美匹配树等图的结构得到刻画.     

Degree Condition for Subdivisions of Unicyclic Graphs

If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.     

Spanning k-Trees of n-Connected Graphs

A tree is called a k-tree if the maximum degree is at most k. We prove the following theorem, by which a closure concept for spanning k-trees of n-connected graphs can be defined. Let k ≥ 2 and n ≥ 1 be integers, and let u and v be a pair of nonadjacent vertices of an n-connected graph G such that deg _G (u) + deg _G (v) ≥ |G| − 1 − (k − 2)n, where |G| denotes the order of G. Then G has a spanning k-tree if and only if G + uv has a spanning k-tree.     

Connected Factors and Spanning Trees in Graphs

 Let G be a graph, and g, f, f′ be positive integer-valued functions defined on V(G). If an f′-factor of G is a spanning tree, we say that it is f′-tree. In this paper, it is shown that G contains a connected (g, f+f′−1)-factor if G has a (g, f)-factor and an f′-tree. Received: October 30, 2000 Final version received: August 20, 2002     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

Even Cycles in Graphs with Many Odd Cycles

It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn ^1/6 even cycles of different lengths. Other results on the number of even cycle lengths which appear in graphs with many different odd length cycles will be given. Received: October 15, 1997     

Vertex-Disjoint Triangles in Claw-Free Graphs with Minimum Degree at Least Three

. Our main result is as follows: For any integer , if G is a claw-free graph of order at least and with minimum degree at least 3, then G contains k vertex-disjoint triangles unless G is of order and G belongs to a known class of graphs. We also construct a claw-free graph with minimum degree 3 on n vertices for each such that it does not contain k vertex-disjoint triangles. We put forward a conjecture on vertex-disjoint triangles in -free graphs. Received: November 21, 1996/Revised: Revised February 19, 1998     

Consistent Cycles in Graphs and Digraphs

Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle of Γ is called G-consistent whenever there is an element of G whose restriction to is the 1-step rotation of . Consistent cycles in finite arc-transitive graphs were introduced by J. H. Conway in his public lectures at the Second British Combinatorial Conference in 1971. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general setting of arbitrary groups of automorphisms of graphs and digraphs. Supported in part by ``Ministrstvo za šolstvo, znanost in šport Republike Slovenije', bilateral project BI-USA/04-05/38.     

Word-Representability of Face Subdivisions of Triangular Grid Graphs

A graph \(G=(V,E)\) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if \((x,y)\in E\). A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph \(K_4\). Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpiński gasket graph is word-representable.     

Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let Forb*(H) be the class of graphs with no induced subdivision of H. It was conjectured in [J Graph Theory, 24 (1997), 297–311] that, for every graph H, there is a function f_H: ?→? such that for every graph G∈Forb*(H), χ(G)≤f_H(ω(G)). We prove this conjecture for several graphs H, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex‐disjoint pendant edges), and what we call a “necklace,” that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:49–68, 2012