A note on vertex pancyclic oriented graphs

Let D be an oriented graph of order n ≥ 9, minimum degree at least n − 2, such that, for the choice of distinct vertices x and y, either xy ∈ E(D) or d⁺(x) + d⁻(y) ≥ n − 3. Song (J. Graph Theory 18 (1994), 461–468) proved that D is pancyclic. In this note, we give a short proof, based on Song's result, that D is, in fact, vertex pancyclic. This also generalizes a result of Jackson (J. Graph Theory 5 (1981), 147–157) for the existence of a hamiltonian cycle in oriented graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 313–318, 1999     

Divisibility and structure constraints on automorphism groups of almost perfect 1-factorizations of

Let G be a permutation group acting on a set with N elements such that every permutation with more than m fixed points is the identity. It is easy to verify that |G| divides N(N − 1) ··· (N − m). We show that if gcd(|G|, m!) = 1, then |G| divides (N − i)(N − j) for some i and j satisfying 0 ≤ i < j ≤ m. We use this to show that any almost perfect 1-factorization of K_2n has an automorphism group whose cardinality divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2 as long as n is odd. An almost perfect 1-factorization (or APOF) is a 1-factorization in which the union of any three distinct 1-factors is connected. This result contrasts with an example of an APOF on K₁₂ given by Cameron which has PSL(2, ℤ₁₁) as its automorphism group [with cardinality 12(11)(5)]. When n is even and the automorphism group is solvable, we show that either G acts vertex transitively and n is a power of two, or |G| divides 2n − 2^a for some integer a with 2^a dividing 2n, or else |G| divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2. We also give a number of structure results concerning these automorphism groups. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 355–380, 1998     

From regular boundary graphs to antipodal distance-regular graphs

Let Γ be a regular graph with n vertices, diameter D, and d + 1 different eigenvalues λ > λ₁ > ··· > λ_d. In a previous paper, the authors showed that if P(λ) > n − 1, then D ≤ d − 1, where P is the polynomial of degree d − 1 which takes alternating values ± 1 at λ₁, …, λ_d. The graphs satisfying P(λ) = n − 1, called boundary graphs, have shown to deserve some attention because of their rich structure. This paper is devoted to the study of this case and, as a main result, it is shown that those extremal (D = d) boundary graphs where each vertex have maximum eccentricity are, in fact, 2-antipodal distance-regular graphs. The study is carried out by using a new sequence of orthogonal polynomials, whose special properties are shown to be induced by their intrinsic symmetry. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 123–140, 1998     

Digraphs containing every possible pair of dicycles

Let D = (V, A) be a directed graph of order n ≥ 4. Suppose that the minimum degree of D is at least (3n − 3)/2. Then for any two integers s and t with s ≥ 2, t ≥ 2 and s + t ≤ n, D contains two vertex‐disjoint directed cycles of lengths s and t, respectively. Moreover, the condition on the minimum degree is sharp. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 154–162, 2000     

A linear Vizing‐like relation relating the size and total domination number of a graph

We prove that m ≤ Δ (n ? γ_t) for every graph each component of which has order at least 3 of order n, size m, total domination number γ_t, and maximum degree Δ ≥ 3. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 285–290, 2005     

Locally semicomplete digraphs that are complementary m-pancyclic

If A and B are two subdigraphs of D, then we denote by d_D (A, B) the distance between A and B. Let D be a 2-connected locally semicomplete digraph on n ≥ 6 vertices. If S is a minimum separating set of D and then m = max{3, d + 2} ≤ n/2 and D contains two vertex-disjoint dicycles of lengths t and n − t for every integer t satisfying m ≤ t ≤ n/2, unless D is a member of a family of locally semicomplete digraphs. This result extends those of Reid (Ann. Discrete Math. 27 (1985), 321–334) and Song (J. Combin. Theory B 57 (1993), 18–25) for tournaments, and it confirms two conjectures of Bang-Jensen (Discrete Math. 100 (1992), 243–265. © 1996 John Wiley & Sons, Inc.     

Forbidden induced subgraph characterization of cograph contractions

Let S₁, S₂,…,S_t be pairwise disjoint non‐empty stable sets in a graph H. The graph H^* is obtained from H by: (i) replacing each S_i by a new vertex q_i; (ii) joining each q_i and q_j, 1 ≤ i # j ≤ t, and; (iii) joining q_i to all vertices in H – (S₁ ∪ S₂ ∪ ··· ∪ S_t) which were adjacent to some vertex of S_i. A cograph is a P₄‐free graph. A graph G is called a cograph contraction if there exist a cograph H and pairwise disjoint non‐empty stable sets in H for which G ? H^*. Solving a problem proposed by Le [ 2 ], we give a finite forbidden induced subgraph characterization of cograph contractions. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 217–226, 2004     

An El-Zahár type condition ensuring path-factors

Let G be a connected graph of order n and suppose that n = n₁ + ··· + n_k where n_i ≥ 2 are integers. The following will be proved: If G has minimum degree at least ⌊ ½n₁⌋ + ··· + ⌊ ½n_k⌋ then G has a spanning subgraph which consists of paths of orders n₁, …, n_k. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 39–42, 1998     

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2‐colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α‐size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α‐size of trees with maximum degree three. Let α₃(n) be the smallest α‐size among all trees with n vertices, each of degree at most three. We prove that α₃(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α₃(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

A generalization of a result of Tran Van Trung on t-designs

We prove that if there exists a t − (v, k, λ) design satisfying the inequality for some positive integer j (where m = min{j, v − k} and n = min {i, t}), then there exists a t − (v + j, k, λ ()) design. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 107–112, 1999     

Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

On the diameter of i‐center in a graph without long induced paths

A graph G is said to be P_t‐free if it does not contain an induced path on t vertices. The i‐center C_i(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t/2⌋ ≤ i ≤ t − 2, with the property that, in every connected P_t‐free graph G, the i‐center C_i(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[C_i(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 235–241, 1999     

k-Bounded classes of dominant-independent perfect graphs

Let α(G), γ(G), and i(G) be the independence number, the domination number, and the independent domination number of a graph G, respectively. For any k ≥ 0, we define the following hereditary classes: αi(k) = {G : α(H) − i(H) ≤ k for every H ∈ ISub(G)}; αγ(k) = {G : α(H) − γ(H) ≤ k for every H ∈ ISub(G)}; and iγ(k) = {G : i(H) − γ(H) ≤ k for every H ∈ ISub(G)}, where ISub(G) is the set of all induced subgraphs of a graph G. In this article, we present a finite forbidden induced subgraph characterization for αi(k) and αγ(k) for any k ≥ 0. We conjecture that iγ(k) also has such a characterization. Up to the present, it is known only for iγ(0) (domination perfect graphs [Zverovich & Zverovich, J Graph Theory 20 (1995), 375–395]). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 303–310, 1999     

A lower bound for the crossing number of Cm × Cn

We prove that the crossing number of C_m × C_n is at least (m − 2)n/3, for all m, n such that n ≥ m. This is the best general lower bound known for the crossing number of C_m × C_n, whose exact value has been long conjectured to be (m − 2)n, for 3 ≤ m ≤ n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 222–226, 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     

A lower bound for partial list colorings

Let G be an n-vertex graph with list-chromatic number χ_ℓ. Suppose that each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least t_n/χ_ℓ vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a corollary, we show that at least of the conjectured number can be colored. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 390–393, 1999     

Star-factors of tournaments

Let S_m denote the m-vertex simple digraph formed by m − 1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m − m lg lg m ≤ f(m) ≤ 4m² − 6m for sufficiently large m. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 141–145, 1998