Izak Broere

A survey of hereditary properties of graphs

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

Majority domination in graphs

Abstract A two-valued function f defined on the vertices of a graph G = (V, E), f: V → -1, 1, is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices v ϵ V, f(N[v]) ⩾ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a majority dominating function is f(V) = ∑f(v), over all vertices v ϵ V. The majority domination number of a graph G, denoted γmaj(G), equals the minimum weight of a majority dominating function of G. In this paper we present properties of the majority domination number and establish its value for various classes of graphs. We show that the decision problem corresponding to the problem of computing γmaj(G) is NP-complete.

PRODUCTS OF CIRCULANT GRAPHS

ABSTRACT Graph products of circulants are studied. It is shown that if G and H are circulants and gcd(v(G), v(H)) = 1, then every B-product of G and H is again a circulant. We prove that if m ≠ 2, then the generalised prism K2 mxCn is a circulant iff n is odd. A similar result is deduced for the conjunction. We also prove that Cp x Cq is a circulant iff p and q are relatively prime. We close by showing that the composition of two circulants is again a circulant and explicitly describe the resultant circulants jump sequence in terms of the constituent circulants jump sequences.

THE CLIQUE NUMBERS AND CHROMATIC NUMBERS OF CERTAIN PALEY GRAPHS

Abstract The problem of finding a formula for the clique number of the Paley graph G*(p) where p is a prime number such that p ≡ 1(mod 4) seems to be a hard one. These clique numbers are known for the caws 5 ⋚ p ⋚ 1601. In this paper we show that the clique number and chromatic number of the Paley graph G*(p2r) are both pr, where p is an odd prime and r is a natural number. Upper and lower bounds for the clique numbers and chromatic numbers of some related graphs are also determined.

Maximal graphs with respect to hereditary properties

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P1, . . . ,Pn be properties of graphs. We say that a graph G has property P1◦ · · · ◦Pn if the vertex set of G can be partitioned into n sets V1, . . . , Vn such that the subgraph of G induced by Vi has property Pi; i = 1, . . . , n. A hereditary property R is said to be reducible if there exist two hereditary properties P1 and P2 such that R = P1◦P2. If P is a hereditary property, then a graph G is called Pmaximal if G has property P but G+e does not have property P for every e ∈ E(G). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.

On the order of uniquely (k,m)-colourable graphs

Abstract For integers k ⩾1 and m ⩾2 a ( k,m )-colouring of a graph G is a colouring of the vertices of G in k colours such that no m -clique of G is monocoloured. The m th chromatic number χ m ( G ) of G is the least k for which G has a ( /IT>)-colouring. A graph G is uniquely ( k,m )-colourable if χ m ( G )= k and any two ( k,m )-colourings of G induce the same partition of V(G) . We prove that, for k ⩾2 and m ⩾3, there exists a uniquely ( k,m )-colourable graph of order n if and only if n ⩾ k ( m −1)+ m ( k −1). In the process, we determine the only uniquely (2, m )-colourable graph of order 3 m −2 and describe the structure of all the uniquely ( k,m )-colourable graphs of order k ( m −1)+ m ( k −1).

Minimal reducible bounds for planar graphs

Abstract For properties of graphs P 1 and P 2 a vertex ( P 1 , P 2 ) -partition of a graph G is a partition (V 1 ,V 2 ) of V(G) such that each subgraph G[V i ] induced by V i has property P i , i=1,2 . The class of all vertex ( P 1 , P 2 ) -partitionable graphs is denoted by P 1 ∘ P 2 . An additive hereditary property R is reducible if there exist additive hereditary properties P 1 and P 2 such that R = P 1 ∘ P 2 , otherwise it is irreducible. For a given property P a reducible property R is called a minimal reducible bound for P if P ⊆ R and there is no reducible property R ′ satisfying P ⊆ R ′⊂ R . In this paper we give a survey of known reducible bounds and we prove some new minimal reducible bounds for important classes of planar graphs. The connection between our results and Barnettes conjecture is also presented.

Meet- and join-irreducibility of additive hereditary properties of graphs

An additive hereditary property of graphs is any class of simple graphs which is closed under unions, subgraphs and isomorphisms. The set of all such properties is a lattice with set inclusion as the partial ordering. We study the elements of this lattice which are meet- join- and doubly-irreducible. The significance of these elements for the lattice of ideals of this lattice is discussed.

A Construction of Uniquely C4-free colourable Graphs

Abstract An F-free colouring of a graph G is a partition {V1,V2,…,Vn} of the vertex set V(G) of G such that F is not an induced subgraph of G[Vi] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C4-free colourable graphs exist.

Critically cochromatic graphs

A graph G is critically n-cochromatic if (its cochromatic number) z(G) = n and z(G - v) = n - 1 for every vertex v of G. Properties of critically n-cochromatic graphs are discussed and we also construct graphs that are critically n-chromatic and critically n-cochromatic.