Cycles containing all vertices of maximum degree

For a graph G and an integer k, denote by V_k the set {v ∈ V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |V_k| ≤ k, then G has a cycle containing all vertices of V_k. It is shown that the upper bound k on |V_k| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |V_k| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented. © 1993 John Wiley & Sons, Inc.     

On Enomoto’s problems in a bipartite graph

In this paper, we obtain the following result: Let k, n ₁ and n ₂ be three positive integers, and let G = (V ₁,V ₂;E) be a bipartite graph with |V1| = n ₁ and |V ₂| = n ₂ such that n ₁ ⩾ 2k + 1, n ₂ ⩾ 2k + 1 and |n ₁ − n ₂| ⩽ 1. If d(x) + d(y) ⩾ 2k + 2 for every x ∈ V ₁ and y ∈ V ₂ with xy $ \notin $ \notin E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.     

The symmetric (2k,k)‐graphs

A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K_{2k + 2} − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

The average Steiner distance of a graph

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μ_n(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μ_n(G) ≤ μ_k(G) + μ_n+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μ_n(T) ≤ (n/k)μ_k(T) and the range for μ_n(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.     

Degree bounds for the circumference of 3‐connected graphs

Let C be a longest cycle in the 3‐connected graph G and let H be a component of G ? V(C) such that |V(H)| ≥ 3. We supply estimates of the form |C| ≥ 2d(u) + 2d(v) ? α(4 ≤ α ≤ 8), where u,v are suitably chosen non‐adjacent vertices in G. Also the exceptional classes for α = 6,7,8 are characterized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

Induced matching extendable graphs

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K₂, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are C_n × K₂, for n ≥ 3, and C_2n(1, n), for n ≥ 2, where C_2n(1, n) is the graph with 2n vertices x₀, x₁, …, x_2n−1, such that x_ix_j is an edge of C_2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998     

Covering the vertices of a graph by cycles of prescribed length

The main theorem of that paper is the following: let G be a graph of order n, of size at least (n² - 3n + 6)/2. For any integers k, n₁, n₂,…,n_k such that n = n₁ + n₂ +. + n_k and n_i ? 3, there exists a covering of the vertices of G by disjoint cycles (C_i) =l…k with |C_i| = ni, except when n = 6, n₁ = 3, n₂ = 3, and G is isomorphic to G₁, the complement of G₁ consisting of a C₃ and a stable set of three vertices, or when n = 9, n₁ = n₂ = n₃ = 3, and G is isomorphic to G₂, the complement of G₂ consisting of a complete graph on four vertices and a stable set of five vertices. We prove an analogous theorem for bipartite graphs: let G be a bipartite balanced graph of order 2n, of size at least n² - n + 2. For any integers s, n₁, n₂,…,n_s with n_i ? 2 and n = n₁ + n₂ + ? + n_s, there exists a covering of the vertices of G by s disjoint cycles C_i, with |C_i| = 2n_i.     

On Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

Cycles in bipartite graphs and an application in number theory

Let G = G(A, B) be a bipartite graph with |A| = u, |B| = v, and let / be a positive integer. In this paper we prove the following result: If v ≤ u, uv ≤ n, m = |E(G)|, and m ≥ max{180/u, 20/n^{1/2(1+(1/l))},} then G contains a C_2/. © 1995 John Wiley & Sons, Inc.     

On low bound of degree sequences of spanning trees in K-edge-connected graphs

A graph G = (V, E) is k-edge-connected if for any subset E′ ⊆ E,|E′| < k, G − E′ is connected. A d_k-tree T of a connected graph G = (V, E) is a spanning tree satisfying that ∀v ∈ V, d_T(v) ≤ + α, where [·] is a lower integer form and α depends on k. We show that every k-edge-connected graph with k ≥ 2, has a d_k-tree, and α = 1 for k = 2, α = 2 for k ≥ 3. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 87–95, 1998     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

On a Conjecture on Directed Cycles in a Directed Bipartite Graph

Let D = (V ₁, V ₂; A) be a directed bipartite graph with |V ₁| = |V ₂| = n ≥ 2. Suppose that d _D (x) + d _D (y) ≥ 3n for all x ∈ V ₁ and y ∈ V ₂. Then, with one exception, D contains two vertex-disjoint directed cycles of lengths 2n ₁ and 2n ₂, respectively, for any positive integer partition n = n ₁ + n ₂. This proves a conjecture proposed in [9].     

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     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.