Marietjie Frick

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.

A Survey of (m, k)-Colorings

Abstract For a given graph invariant γ an (m, k)γ -coloring of a graph G is a partition of the vertex set of G into m subsets V1,…, V1 such that γ( ) ≤kfor i = 1,…, m. Various aspects of (m, k)γ-colorings are compared for the cases where γ is taken to be, in turn, the clique number, the maximum degree, the degeneracy and the path number.

Extremal results on defective colorings of graphs

Abstract A graph is (m,k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. The k-defective chromatic number, χk(G), of a graph G is the minimum m for which G is (m,k)-colorable. Among other results, we prove that the smallest orders among all uniquely (m,k)-colorable graphs and all minimal (m,k)-chromatic graphs are m(k+2) − 1 and (m − 1)(k + 1)+1, respectively, and we determine all the extremal graphs in the latter case. We also obtain a necessary condition for a sequence to be a defective chromatic number sequence χ0(G), χ1(G), χ2(G),…; it is an open question whether this condition is also sufficient.

The directed path partition conjecture

The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D〈A〉 has more than a vertices and no path in D〈B〉 has more than b vertices.We develop methods for finding the desired partitions for various classes of digraphs.

The Path Partition Conjecture is true for claw-free graphs

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that @t(G)=a+b, then the vertex set of G has a partition (A,B) such that @t()=)=

A path(ological) partition problem

Let τ(G) denote the number of vertices in a longest path of the graph G and let k1 and k2 be positive integers such that τ(G) = k1+k2. The question at hand is whether the vertex set V (G) can be partitioned into two subsets V1 and V2 such that τ(G[V1]) ≤ k1 and τ(G[V2]) ≤ k2. We show that several classes of graphs have this partition property.

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.

Uniquely partitionable graphs

Let P1, . . . ,Pn be properties of graphs. A (P1, . . . ,Pn)-partition of a graph G is a partition of the vertex set V (G) into subsets V1, . . . , Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1, . . . , n. A graph G is said to be uniquely (P1, . . . ,Pn)-partitionable if G has exactly one (P1, . . . ,Pn)-partition. A property P is called hereditary if every subgraph of every graph with property P also has property P. If every graph that is a disjoint union of two graphs that have property P also has property P, then we say that P is additive. A property P is called degenerate if there exists a bipartite graph that does not have property P. In this paper, we prove that if P1, . . . ,Pn are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (P1, . . . ,Pn)-partitionable graph.

Smallest claw-free, 2-connected, nontraceable graphs and the construction of maximal nontraceable graphs

We determine the smallest claw-free, 2-connected, nontraceable graphs and use one of these graphs to construct a new family of 2-connected, claw-free, maximal nontraceable graphs.

Detour chromatic numbers

The nth detour chromatic number, χn(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ (G) . We conjecture that χn(G) ≤ d τ(G) n e for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.