Directed star decompositions of the complete directed graph

An (s, t)-directed star is a directed graph with s + t + 1 vertices and s + t arcs; s vertices have indegree zero and outdegree one, t have indegree one and outdegree zero, and one has indegree s and outdegree t. An (s, t)-directed star decomposition is a partition of the arcs of a complete directed graph of order n into (s, t)-directed starsx. We establish necessary and sufficient conditions on s, t, and n for an (s, t)-directed star decomposition of order n to exist.     

The star chromatic number of a graph

We study a generalization of the notion of the chromatic number of a graph in which the colors assigned to adjacent vertices are required to be, in a certain sense, far apart. © 1993 John Wiley & Sons, Inc.     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

The splitting number of the complete graph

If a given graphG can be obtained bys vertex identifications from a suitable planar graph ands is the minimum number for which this is possible thens is called the splitting number ofG. Here a formula for the splitting number of the complete graph is derived.     

Acyclic graph coloring and the complexity of the star chromatic number

Star chromatic number, introduced by A. Vince, is a natural generalization of chromatic number. We consider the question, “When is χ* < χ?” We show that χ* < χ if and only if a particular digraph is acyclic and that the decisioin problem associated with this question is probably not in NP though it is both NP-hard and NP-easy. © 1993 John Wiley & Sons, Inc.     

Packing almost stars into the complete graph

We verify that the Tree Packing Conjecture is true for all sequences of trees T₁, …, T_n such that there exists χ_i E V(T_i) and T_i − χ_i has at least isolated vertices. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 169–172, 1997     

The interval number of a complete multipartite graph

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line $\text{R}$. Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is $\text{?}\text{(mn + 1)}\text{(m + n)}\text{?}$. Matthews showed that the interval number of the complete multipartite graph K(n₁,n₂,…,n_p) was the same as the interval number of K(n₁,n₂) when n₁ = n₂ = ? = n_p. Trotter and Hopkins showed that i(K(n₁,n₂,…,n_p)) ≤ 1 + i(K(n₁,n₂)) whenever p ≥ 2 and n₁≥n₂≥ ? ≥n_p. West showed that for each n ≥ 3, there exists a constant c_n so that if p ≥ c_n,n₁ = n²?n ?1, and n₂ = n₃ = ? n_p = n, then i(K(n₁,n₂,…,n_p) = 1 + i(K(n₁, n₂)). In view of these results, it is natural to consider the problem of determining those pairs (n₁,n₂) with n₁ ≥ n₂ so that i(K(n₂,…,n_p)) = i(K(n₁,n₂)) whenever p ≥ 2 and n₂ ≥ n₃ ≥ ? ≥ n_p. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n² ? n ? 1, n) for n ≥ 3 and (7,5).     

Star number and star arboricity of a complete multigraph

Let G be a multigraph. The star number s(G) of G is the minimum number of stars needed to decompose the edges of G. The star arboricity sa(G) of G is the minimum number of star forests needed to decompose the edges of G. As usual λK _n denote the λ-fold complete graph on n vertices (i.e., the multigraph on n vertices such that there are λ edges between every pair of vertices). In this paper, we prove that for n ⩾ 2

The splitting number of the complete graph in the projective plane

If a given graphG can be obtained bys vertex identifications from a suitable graph embeddable in the projective plane ands is the minimum number for which this is possible thens is called the splitting number ofG in the projective plane. Here a formula for the splitting number of the complete graph in the projective plane is derived.     

Chromatic number and complete graph substructures for degree sequences

Given a graphic degree sequence D, let χ(D) (respectively ω(D), h(D), and H(D)) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is D. It is proved that χ(D)≤h(D). Moreover, it is shown that a subdivision of a clique of order χ(D) exists where each edge is subdivided at most once and the set of all subdivided edges forms a collection of disjoint stars. This bound is an analogue of the Hajós Conjecture for degree sequences and, in particular, settles a conjecture of Neil Robertson that degree sequences satisfy the bound χ(D) ≤ H(D) (which is related to the Hadwiger Conjecture). It is also proved that χ(D) ≤ 6/5 ω(D)+ 3/5 and that χ(D) ≤ 4/5 ω(D) + 1/5 Δ(D)+1, where Δ(D) denotes the maximum degree in D. The latter inequality is related to a conjecture of Bruce Reed bounding the chromatic number by a convex combination of the clique number and the maximum degree. All derived inequalities are best possible     

完全3-部图K_(1，10，n)的交叉数

在上世纪五十年代初,Zarankiewicz猜想完全2-部图Km,m(m≤n)的交叉数为[m/2][m-1/2][n/2][n-1/2](对任意实数x,[x]表示不超过x的最大整数),目前只证明了当m ≤ 6时,Zarankiewicz猜想是正确的.假定Zarankiewicz猜想对m=11的情形成立,本文确定完全3-部图K1,10,n的交叉数.     

Decomposing the complete graph into cycles of many lengths

For all oddv 3 the complete graph onvK _v vertices can be decomposed intov – 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v – 1. Also, for all oddv 7,K _v can be decomposed intov – 3 edge disjoint cycles whose lengths are 3, 4,...,v – 4,v – 2,v – 1,v. Research supported by Australian Research Council grant A49130102     

Isomorphic factorization of the complete graph into Cayley digraphs

Let Z_p denote the cyclic group of order p where p is a prime number. Let X = X(Z_p, H) denote the Cayley digraph of Z_p with respect to the symbol H. We obtain a necessary and sufficient condition on H so that the complete graph on p vertices can be edge‐partitioned into three copies of Cayley digraphs of the same group Z_p each isomorphic to X. Based on this condition on H, we then enumerate all such Cayley graphs and digraphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 243–256, 2006     

Decomposition of the complete directed graph into k-circuits

We study the decomposition of $\text{K}{}_{\mathrm{n}}{}^1$ (the complete directed graph with n vertices) into arc-disjoint elementary k-circuits, primarily for the case k even. We solve the problem for many values of (n, k) and in particular for all n in the cases k = 4, 6, 8, and 16.     

Minimum number of below average triangles in a weighted complete graph

Let G be an edge weighted graph with n nodes, and let A(3,G) be the average weight of a triangle in G. We show that the number of triangles with weight at most equal to A(3,G) is at least (n−2) and that this bound is sharp for all n≥7. Extensions of this result to cliques of cardinality k>3 are also discussed.     

On cyclic decompositions of the complete graph into the 2-regular graphs

The symbol C(m₁ ⁿ ₁m₂ ⁿ ₂...m_s ⁿ _s) denotes a 2-regular graph consisting ofn _i cycles of lengthm _i , i=1, 2,…,s. In this paper, we give some construction methods of cyclic(K _v ,G)-designs, and prove that there exists a cyclic(K _v , G)-design whenG=C((4m ₁) ⁿ ₁(4m ₂) ⁿ ₂...(4m _s ) ⁿ _s andv ≡ 1 (mod 2¦G¦).     

Packings of the complete directed graph with m-circuits

A packing of the complete directed symmetric graph DKv with m-circuits, denoted by(v,m)-DCP, is defined to he a family of are-disjoint m-circuits of DK, such that any one arc of DKv occurs in at most one m circuit. The packing number P(v,m) is the maximum number of m-circults in such a packing. The packing problem is to determine the value P(v,m) for everyinteger v ≥ m. In this paper, the problem is reduced to the case m 6 ≤v≤2m-[(4m-3的平方极) 1/2],for any fixed even integer m≥4,In particular,the values of P（v,m） are completely determined for m=12,14 and 16.