On factorable degree sequences

We call a degree sequence graphic (respectively, k-factorable, connected k-factorable) if there exists a graph (respectively, a graph with a k-factor, a graph with a connected k-factor) with the given degree sequence. In this paper we give a necessary and sufficient condition for a k-factorable sequence to be connected k-factorable when k ? 2. We also prove that every k-factorable sequence is (k − 2) factorable and 2-factorable, and also 1-factorable, when the sequence is of even length. Some conjectures are stated and it is also proved that, if {d_i} and {d_i − k} are graphic, then {d_i − r} is graphic for 0 ≤ r ≤ k provided rn is even.     

On 2-factors with k components

In this paper we study the minimum degree condition for a Hamiltonian graph to have a 2-factor with k components. By proving a conjecture of Faudree et al. [A note on 2-factors with two components, Discrete Math. 300 (2005) 218-224] we show the following. There exists a real number ε>0 such that for every integer k?2 there exists an integer n₀=n₀(k) such that every Hamiltonian graph G of order n?n₀ with has a 2-factor with k components.     

On the 1-factors of n-connected graphs

L. W. Beineke and M. D. Plummer have recently proved [1] that every n-connected graph with a 1-factor has at least n different 1-factors. The main purpose of this paper is to prove that every n-connected graph with a 1-factor has at least as many as n(n − 2)(n − 4) … 4 · 2, (or: n(n − 2)(n − 4) … 5 · 3) 1-factors. The main lemma used is: if a 2-connected graph G has a 1-factor, then G contains a vertex V (and even two such vertices), such that each edge of G, incident to V, belongs to some 1-factor of G.     

A note on domination, girth and minimum degree

Let G be a graph of order n, minimum degree δ?2, girth g?5 and domination number γ. In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269-275] proved that γ?⌈n/2-g/6⌉. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131-141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195-212] who for i∈{1,2} determined a finite set of graphs G_i such that γ?⌈n/2-g/6-(3i+3)/6⌉ unless G is a cycle or G∈G_i.Our main result is that for every i∈N there is a finite set of graphs G_i such that γ?n/2-g/6-i unless G is a cycle or G∈G_i. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.     

Caterpillar arboricity of planar graphs

We solve a conjecture of Roditty, Shoham and Yuster [P.J. Cameron (Ed.), Problems from the 17th British Combinatorial Conference, Discrete Math., 231 (2001) 469-478; Y. Roditty, B. Shoham, R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411-417] on the caterpillar arboricity of planar graphs. We prove that for every planar graph G=(V,E), the edge set E can be partitioned into four subsets (E_i)_1?i?4 in such a way that G[E_i], for 1?i?4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.     

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1,2]-factor F_G, i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α(G)=α(F_G) for some perfect [1,2]-factor F_G. This class contains all well-covered graphs G without isolated vertices of order n with α?(n-1)/2, and in particular all very well-covered graphs.     

Cayley图的泛因子性质

一个κ-正则图若满足对任意正整数s,1≤s≤κ,均存在一个s-因子或一个2[s/2]因子,则称其有泛因子或偶泛因子性质.本文证明了每个奇度Cayley图是泛因子的,每个偶度Carley图是偶泛因子的.同时证明了二面体群上的每个Cayley图均是泛因子的.     

Eccentric sequences and eccentric sets in graphs

By a graph we mean a finite undirected connected graph of order p, p ? 2, with no loops or multiple edges. A finite non-decreasing sequence S: s₁, s₂, …, s_p, p ? 2, of positive integers is an eccentric sequence if there exists a graph G with vertex set V(G) = {v₁, v₂, …, v_p} such that for each i, 1 ? i ? p, s, is the eccentricity of v₁. A set S is an eccentric set if there exists a graph G such that the eccentricity ρ(v₁) is in S for every v₁ ? V(G), and every element of S is the eccentricity of some vertex in G. In this note we characterize eccentric sets, and we find the minimum order among all graphs whose eccentric set is a given set, to obtain a new necessary condition for a sequence to be eccentric. We also present some properties of graphs having preassigned eccentric sequences.     

On connected factors inK 1,3-free graphs

A graph is said to beK _1,3-free if it contains noK _1,3 as an induced subgraph. It is shown in this paper that every 2-connectedK _1,3-free graph contains a connected [2,3]-factor. We also obtain that every connectedK _1,3-free graph has a spanning tree with maximum degree at most 3. This research is partially supported by the National Natural Sciences Foundation of China and by the Natural Sciences Foundation of Shandong Province of China.     

Linkability in iterated line graphs

We prove that for every graph H with the minimum degree δ?5, the third iterated line graph L³(H) of H contains as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number k_G such that for every i?k_G the i-iterated line graph of G is -linked. Since the degree of Lⁱ(G) is even, the result is best possible.     

Vertex pancyclicity in quasi claw-free graphs

This paper generalizes the concept of locally connected graphs. A graph G is triangularly connected if for every pair of edges e₁,e₂∈E(G), G has a sequence of 3-cycles C₁,C₂,…,C_l such that e₁∈C₁,e₂∈C_l and E(C_i)∩E(C_i+1)≠∅ for 1?i?l-1. In this paper, we show that every triangularly connected quasi claw-free graph on at least three vertices is vertex pancyclic. Therefore, the conjecture proposed by Ainouche is solved.     

A bijection between certain non-crossing partitions and sequences

We present a bijection between non-crossing partitions of the set [2n+1] into n+1 blocks such that no block contains two consecutive integers, and the set of sequences such that 1?s_i?i, and if s_i=j, then s_i-r?j-r for 1?r?j-1.     

A pair of forbidden subgraphs and perfect matchings in graphs of high connectivity

Sumner [7] proved that every connected K _1,3-free graph of even order has a perfect matching. He also considered graphs of higher connectivity and proved that if m ≥ 2, every m-connected K _1,m+1-free graph of even order has a perfect matching. In [6], two of the present authors obtained a converse of sorts to Sumner’s result by asking what single graph one can forbid to force the existence of a perfect matching in an m-connected graph of even order and proved that a star is the only possibility. In [2], Fujita et al. extended this work by considering pairs of forbidden subgraphs which force the existence of a perfect matching in a connected graph of even order. But they did not settle the same problem for graphs of higher connectivity. In this paper, we give an answer to this problem. Together with the result in [2], a complete characterization of the pairs is given.     

Disjoint triangles and quadrilaterals in a graph

Let G be a simple graph of order n and s and k be two positive integers. Brandt et al. obtained the following result: If s?k, n?3s+4(k-s) and σ₂(G)?n+s, then G contains k disjoint cycles C₁,…,C_k satisfying |C_i|=3 for 1?i?s and |C_i|?4 for s<i?k. In the above result, the length of C_i is not specified for s<i?k. We get a result specifying the length of C_i for each s<i?k if n?3s+4(k-s)+3.     

关于K_（1，r）－free图的连通因子

本文证明了每个连通的Ｋ１，ｒ－ｆｒｅｅ图Ｇ，如果有［ｆ，ｇ］－因子Ｆ，则它就有包含Ｆ的［ｆ，ｇ＋ｒ－１］连通因子．     

A construction of Gray codes inducing complete graphs

A binary Gray code G(n) of length n, is a list of all 2ⁿn-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in G(n), denoted by S(n)?s₁,s₂,…,s_2ⁿ-1, is called the transition sequence of the Gray code G(n). The graph G_G(n) induced by a Gray code G(n) has vertex set {1,2,…,n} and edge set {{s_i,s_i+1}:1?i?2ⁿ-2}. If the first and the last codeword differ only in position s_2ⁿ, the code is cyclic and we extend the graph by two more edges {s_2ⁿ-1,s_2ⁿ} and {s_2ⁿ,s₁}. We solve a problem of Wilmer and Ernst [Graphs induced by Gray codes, Discrete Math. 257 (2002) 585-598] about a construction of an n-bit Gray code inducing the complete graph K_n. The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth Functions and the Problem of their Realization by Two-Terminal Graphs, Akadémiai Kiadó, Budapest, 1968], and which is presented in D.E. Knuth [The Art of Computer Programming, vol. 4, Addison-Wesley as part of “fascicle” 2, USA, 2005].     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

Path factors and parallel knock-out schemes of almost claw-free graphs

An H₁,{H₂}-factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H₁ and all other components (if there are any) isomorphic to the graph H₂. We completely characterise the class of connected almost claw-free graphs that have a P₇,{P₂}-factor, where P₇ and P₂ denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation, we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost claw-free graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n^5.376) time algorithm for claw-free graphs on n vertices).     

On covering vertices of a graph by trees

The purpose of this paper is to initiate study of the following problem: Let G be a graph, and k?1. Determine the minimum number s of trees T₁,…,T_s, Δ(T_i)?k,i=1,…,s, covering all vertices of G. We conjecture: Let G be a connected graph, and k?2. Then the vertices of G can be covered by edge-disjoint trees of maximum degree ?k. As a support for the conjecture we prove the statement for some values of δ and k.