Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Subdivisions of large complete bipartite graphs and long induced paths in k‐connected graphs

It is proved that for every positive integers k, r and s there exists an integer n = n(k,r,s) such that every k‐connected graph of order at least n contains either an induced path of length s or a subdivision of the complete bipartite graph K_k,r. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 270–274, 2004     

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ? 2I and κ k = δ if D ? 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.     

On Moore bipartite digraphs

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore‐like bound in terms of its diameter k and the maximum out‐degrees (d₁, d₂) of its partite sets of vertices. It has been proved that, when d₁d₂ > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The hamiltonian property of the consecutive-3 digraph

A consecutive-ifd digraph is a digraph G(d, n, q, r) whose n nodes are labeled by the residues modulo n and a link from node i to node j exists if and only if j  qi + k (mod n) for some k with r ≤ k ≤ r + d − 1. Consecutive-d digraphs are used as models for many computer networks and multiprocessor systems, in which the existence of a Hamiltonian circuit is important. Conditions for a consecutive-d graph to have a Hamiltonian circuit were known except for gcd(n, d) = 1 and d = 3 or 4. It was conjectured by Du, Hsu, and Hwang that a consecutive-3 digraph is Hamiltonian. This paper produces several infinite classes of consecutive-3 digraphs which are not (respectively, are) Hamiltonian, thus suggesting that the conjecture needs modification.     

Pancyclic graphs and linear forests

Given integers k,s,t with 0≤s≤t and k≥0, a (k,t,s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k,t)-linear forest. A graph G of order n≥k+t is (k,t)-hamiltonian if for any (k,t)-linear forest F there is a hamiltonian cycle containing F. More generally, given integers m and n with k+t≤m≤n, a graph G of order n is (k,t,s,m)-pancyclic if for any (k,t,s)-linear forest F and for each integer r with m≤r≤n, there is a cycle of length r containing the linear forest F. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply that a graph is (k,t,s,m)-pancyclic (or just (k,t,m)-pancyclic) are proved.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

Exponents of a class of two-colored digraphs

A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n???2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.     

Kings in semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete p‐partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete p‐partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r‐king is a vertex q such that every vertex in D can be reached from q by a path of length at most r. Strengthening a theorem by K. M. Koh and B. P. Tan (Discr Math 147 (1995), 171–183) on the number of 4‐kings in multipartite tournaments, we characterize semicomplete multipartite digraphs, which have exactly k 4‐kings for every k = 1, 2, 3, 4, 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 177‐183, 2000     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003     

On the chromatic number of the general Kneser-graph

Given integers k, n, 2 < k < n, let us define a graph with vertex set V = {F ?{1, 2, …, n}: ∩F = k}, and (F, F') is an edge if |F ∩ F′| ≤ 1. We show that for n > n₀(k) the chromatic number of this graph is (k - 1)() + rs, where n = (k - 1)s + r, 0 ≤ r < k - 1.     

An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence s₁,…,s_k,t₁,…,t_k of distinct vertices there exist disjoint paths P₁,…,P_k such that the ends of P_i are s_i and t_i. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.     

Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

On Hamiltonicity of circulant digraphs of outdegree three

This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form {a,ka,c}, where k is an integer. In particular, we prove that if k=−1 or k=2 such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set {3,6,4}.     

The extremal function for 3-linked graphs

A graph is k-linked if for every set of 2k distinct vertices {s₁,…,s_k,t₁,…,t_k} there exist disjoint paths P₁,…,P_k such that the endpoints of P_i are s_i and t_i. We prove every 6-connected graph on n vertices with 5n−14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5n−15 edges that are not 3-linked for arbitrarily large values of n.     

On H‐immersions

For a fixed multigraph H, possibly containing loops, with V(H) = {h₁,…, h_k}, we say a graph G is H‐linked if for every choice of k vertices v₁,…,v_k in G, there exists a subdivision of H in G such that v_i represents h_i (for all i). An H‐immersion in G is similar except that the paths in G, playing the role of the edges of H, are only required to be edge disjoint. In this article, we extend the notion of an H‐linked graph by determining minimum degree conditions for a graph G to contain an H‐immersion with a bounded number of vertex repetitions on any choice of k vertices. In particular, we extend results found in [2,3,5]. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 245–254, 2008     

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004