Marietjie Frick - Profile on Academia.edu (original) (raw)

Papers by Marietjie Frick

The Path Partition Conjecture Holds for a = 9

A Survey of (m, k)-Colorings

Annals of discrete mathematics, 1993

Abstract For a given graph invariant γ an (m, k)γ -coloring of a graph G is a partition of the ve... more 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.

Discrete Mathematics, Dec 1, 1996

For a graph G, the path number r(G) is defined as the order of a longest path in G. An (m, k)~-co... more For a graph G, the path number r(G) is defined as the order of a longest path in G. An (m, k)~-colouring of a graph H is a partition of the vertex set of H into m subsets such that each subset induces a subgraph of H for which r is at most k. The k -z-chromatic number z~(H) is the least m for which H has an (m, kf-colouring. A graph H is uniquely (m, kf-colourable if z~(H) = m and there is only one partition of the vertex set of H which is an (m, k)~-colouring of H. A graph G is called k -r-saturated if r(G) ~ k and z(G + e) ~> k + 1 for all e e E(G). For k = 1, the graphs obtained by taking the join of k -z-saturated graphs (which are empty graphs in this case) are known to be uniquely colourable graphs. In this paper we construct uniquely (m, k)~-colourable graphs (for all positive integers m and k) using k -r-saturated graphs in a similar fashion. As a corollary we characterise those p for which there exists a uniquely (m, kfcolourable graph of order p.

arXiv (Cornell University), Jul 16, 2004

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Ka... more Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ ⌈ 3n-2 2 ⌉ -2 for n ≥ 20 and g(n) ≤ ⌈ 3n-2 2 ⌉ for n ≥ 54 as well as for n ∈ I

The Journal of the Australian Mathematical Society, Nov 1, 1973

arXiv (Cornell University), Jul 16, 2004

We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, a... more We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, and present a construction of an infinite family of maximal nontraceable graphs that realize this bound.

Discussiones Mathematicae Graph Theory, 2013

The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2 ) any pair of po... more The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2 ) any pair of positive integers such that G has no path with more than λ 1 + λ 2 vertices, then there exists a partition (V 1 , V 2 ) of the vertex set of G such that V i has no path with more than λ i vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the PPC that have appeared in the literature since its first formulation in 1981.

Electronic Journal of Combinatorics, Jan 21, 2008

A graph G is maximal nontraceable (MNT) if G does not have a hamiltonian path but, for every e ∈ ... more A graph G is maximal nontraceable (MNT) if G does not have a hamiltonian path but, for every e ∈ E G , the graph G + e has a hamiltonian path. A graph G is 1-tough if for every vertex cut S of G the number of components of G − S is at most |S|. We investigate the structure of MNT graphs that are not 1-tough. Our results enable us to construct several interesting new classes of MNT graphs.

Discrete Mathematics, Nov 1, 2017

An edge-coloured graph G is called properly connected if any two vertices are connected by a path... more An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. Our main result is the following: Let G be a connected graph of order n and k ≥ 2. If |E(G)| ≥ (n−k−1 2) + k + 2, then pc(G) ≤ k except when k = 2 and

Electronic Journal of Combinatorics, Nov 13, 2015

A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is tra... more A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so k-traceable oriented graphs may be regarded as generalized tournaments. It is well-known that all tournaments are traceable. We denote by t(k) the smallest integer bigger than or equal to k such that every k-traceable oriented graph of order at least t(k) is traceable. The Traceability Conjecture states that t(k) 2k − 1 for every k 2 [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008]. We show that for k 2, every k-traceable oriented graph with independence number 2 and order at least 4k − 12 is traceable. This is the last open case in giving an upper bound for t(k) that is linear in k.

Graphs and Combinatorics, Jun 8, 2017

For a given graph property P, we say a graph G is locally P if for each v ∈ V (G), the subgraph i... more For a given graph property P, we say a graph G is locally P if for each v ∈ V (G), the subgraph induced by the open neighbourhood of v has property P. A closed locally P graph is defined analogously in terms of closed neighbourhoods. It is known that connected locally hamiltonian graphs are not necessarily hamiltonian.

Electronic Journal of Combinatorics, Sep 29, 2005

The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partitio... more The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partition of the vertex set of G into two sets, A and B, such that τ (A) ≤ a and τ (B) ≤ b is called an (a, b)-partition of G. If G has an (a, b)-partition for every pair (a, b) of positive integers such that a + b = τ (G), then we say that G is τ-partitionable. The Path Partition Conjecture (PPC), which was discussed by Lovász and Mihók in 1981 in Szeged, is that every graph is τ-partitionable. It is known that a graph G of order n and detour order τ = n − p is τ-partitionable if p = 0, 1. We show that this is also true for p = 2, 3, and for all p ≥ 4 provided that n ≥ p(10p − 3).

Discussiones Mathematicae Graph Theory, 2001

The nth detour chromatic number, χ n (G) of a graph G is the minimum number of colours required t... more The nth detour chromatic number, χ n (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 χ n (G) ≤ τ (G) n 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.

The order of hypotraceable oriented graphs

Discrete Mathematics, Jul 1, 2011

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of or... more A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented

Discrete Mathematics, Jun 1, 1990

For integers k 2 1 and m 2 2 a (k, m)-colouring of a graph G is a colouring of the vertices of G ... more For integers k 2 1 and m 2 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 mth chromatic number x,,,(G) of G is the least k for which G has a (k, m)-colouring. A graph G is uniquely (k, m)-colourable if xm(G) = k and any two (k, m)-colourings of G induce the same partition of V(G). We prove that, for k 2 2 and m 3 3, there exists a uniquely (k, m)-colourable graph of order n if and only if n 2 k(m-1) + m(k-1). In the process, we determine the only uniquely (2, m)-colourable graph of order 3m-2 and describe the structure of all the uniquely (k, m)-colourable graphs of order k(m-1) + m(k-1).

Discrete Mathematics & Theoretical Computer Science, Mar 16, 2017

A digraph is traceable if it has a path that visits every vertex. A digraph D is hypotraceable if... more A digraph is traceable if it has a path that visits every vertex. A digraph D is hypotraceable if D is not traceable but D − v is traceable for every vertex v ∈ V (D). It is known that there exists a planar hypotraceable digraph of order n for every n ≥ 7, but no examples of planar hypotraceable oriented graphs (digraphs without 2-cycles) have yet appeared in the literature. We show that there exists a planar hypotraceable oriented graph of order n for every even n ≥ 10, with the possible exception of n = 14.

Graphs and Combinatorics, Apr 6, 2012

Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We ... more Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We answer his question by presenting an infinite family of planar hypohamiltonian oriented graphs, the smallest of which has order 9. A computer search showed that 9 is the smallest possible order of a hypohamiltonian oriented graph.

Discrete Mathematics, May 1, 2007

We determine the smallest claw-free, 2-connected, nontraceable graphs and use one of these graphs... more 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.

Discussiones Mathematicae Graph Theory, 1997

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every sub... more 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 P 1 ,. .. , P n be properties of graphs. We say that a graph G has property P 1 • • • • •P n if the vertex set of G can be partitioned into n sets V 1 ,. .. , V n such that the subgraph of G induced by V i has property P i ; i = 1,. .. , n. A hereditary property R is said to be reducible if there exist two hereditary properties P 1 and P 2 such that R = P 1 •P 2. If P is a hereditary property, then a graph G is called P-maximal 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.

Electronic Journal of Combinatorics, Jul 19, 2005

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Ka... more Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ 3n−2 2 −2 for n ≥ 20 and g(n) ≤ 3n−2 2

The Path Partition Conjecture Holds for a = 9

A Survey of (m, k)-Colorings

Annals of discrete mathematics, 1993

Abstract For a given graph invariant γ an (m, k)γ -coloring of a graph G is a partition of the ve... more 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.

Discrete Mathematics, Dec 1, 1996

For a graph G, the path number r(G) is defined as the order of a longest path in G. An (m, k)~-co... more For a graph G, the path number r(G) is defined as the order of a longest path in G. An (m, k)~-colouring of a graph H is a partition of the vertex set of H into m subsets such that each subset induces a subgraph of H for which r is at most k. The k -z-chromatic number z~(H) is the least m for which H has an (m, kf-colouring. A graph H is uniquely (m, kf-colourable if z~(H) = m and there is only one partition of the vertex set of H which is an (m, k)~-colouring of H. A graph G is called k -r-saturated if r(G) ~ k and z(G + e) ~> k + 1 for all e e E(G). For k = 1, the graphs obtained by taking the join of k -z-saturated graphs (which are empty graphs in this case) are known to be uniquely colourable graphs. In this paper we construct uniquely (m, k)~-colourable graphs (for all positive integers m and k) using k -r-saturated graphs in a similar fashion. As a corollary we characterise those p for which there exists a uniquely (m, kfcolourable graph of order p.

arXiv (Cornell University), Jul 16, 2004

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Ka... more Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ ⌈ 3n-2 2 ⌉ -2 for n ≥ 20 and g(n) ≤ ⌈ 3n-2 2 ⌉ for n ≥ 54 as well as for n ∈ I

The Journal of the Australian Mathematical Society, Nov 1, 1973

arXiv (Cornell University), Jul 16, 2004

We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, a... more We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, and present a construction of an infinite family of maximal nontraceable graphs that realize this bound.

Discussiones Mathematicae Graph Theory, 2013

The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2 ) any pair of po... more The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2 ) any pair of positive integers such that G has no path with more than λ 1 + λ 2 vertices, then there exists a partition (V 1 , V 2 ) of the vertex set of G such that V i has no path with more than λ i vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the PPC that have appeared in the literature since its first formulation in 1981.

Electronic Journal of Combinatorics, Jan 21, 2008

A graph G is maximal nontraceable (MNT) if G does not have a hamiltonian path but, for every e ∈ ... more A graph G is maximal nontraceable (MNT) if G does not have a hamiltonian path but, for every e ∈ E G , the graph G + e has a hamiltonian path. A graph G is 1-tough if for every vertex cut S of G the number of components of G − S is at most |S|. We investigate the structure of MNT graphs that are not 1-tough. Our results enable us to construct several interesting new classes of MNT graphs.

Discrete Mathematics, Nov 1, 2017

An edge-coloured graph G is called properly connected if any two vertices are connected by a path... more An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. Our main result is the following: Let G be a connected graph of order n and k ≥ 2. If |E(G)| ≥ (n−k−1 2) + k + 2, then pc(G) ≤ k except when k = 2 and

Electronic Journal of Combinatorics, Nov 13, 2015

A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is tra... more A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so k-traceable oriented graphs may be regarded as generalized tournaments. It is well-known that all tournaments are traceable. We denote by t(k) the smallest integer bigger than or equal to k such that every k-traceable oriented graph of order at least t(k) is traceable. The Traceability Conjecture states that t(k) 2k − 1 for every k 2 [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008]. We show that for k 2, every k-traceable oriented graph with independence number 2 and order at least 4k − 12 is traceable. This is the last open case in giving an upper bound for t(k) that is linear in k.

Graphs and Combinatorics, Jun 8, 2017

For a given graph property P, we say a graph G is locally P if for each v ∈ V (G), the subgraph i... more For a given graph property P, we say a graph G is locally P if for each v ∈ V (G), the subgraph induced by the open neighbourhood of v has property P. A closed locally P graph is defined analogously in terms of closed neighbourhoods. It is known that connected locally hamiltonian graphs are not necessarily hamiltonian.

Electronic Journal of Combinatorics, Sep 29, 2005

The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partitio... more The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partition of the vertex set of G into two sets, A and B, such that τ (A) ≤ a and τ (B) ≤ b is called an (a, b)-partition of G. If G has an (a, b)-partition for every pair (a, b) of positive integers such that a + b = τ (G), then we say that G is τ-partitionable. The Path Partition Conjecture (PPC), which was discussed by Lovász and Mihók in 1981 in Szeged, is that every graph is τ-partitionable. It is known that a graph G of order n and detour order τ = n − p is τ-partitionable if p = 0, 1. We show that this is also true for p = 2, 3, and for all p ≥ 4 provided that n ≥ p(10p − 3).

Discussiones Mathematicae Graph Theory, 2001

The nth detour chromatic number, χ n (G) of a graph G is the minimum number of colours required t... more The nth detour chromatic number, χ n (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 χ n (G) ≤ τ (G) n 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.

The order of hypotraceable oriented graphs

Discrete Mathematics, Jul 1, 2011

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of or... more A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented

Discrete Mathematics, Jun 1, 1990

For integers k 2 1 and m 2 2 a (k, m)-colouring of a graph G is a colouring of the vertices of G ... more For integers k 2 1 and m 2 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 mth chromatic number x,,,(G) of G is the least k for which G has a (k, m)-colouring. A graph G is uniquely (k, m)-colourable if xm(G) = k and any two (k, m)-colourings of G induce the same partition of V(G). We prove that, for k 2 2 and m 3 3, there exists a uniquely (k, m)-colourable graph of order n if and only if n 2 k(m-1) + m(k-1). In the process, we determine the only uniquely (2, m)-colourable graph of order 3m-2 and describe the structure of all the uniquely (k, m)-colourable graphs of order k(m-1) + m(k-1).

Discrete Mathematics & Theoretical Computer Science, Mar 16, 2017

A digraph is traceable if it has a path that visits every vertex. A digraph D is hypotraceable if... more A digraph is traceable if it has a path that visits every vertex. A digraph D is hypotraceable if D is not traceable but D − v is traceable for every vertex v ∈ V (D). It is known that there exists a planar hypotraceable digraph of order n for every n ≥ 7, but no examples of planar hypotraceable oriented graphs (digraphs without 2-cycles) have yet appeared in the literature. We show that there exists a planar hypotraceable oriented graph of order n for every even n ≥ 10, with the possible exception of n = 14.

Graphs and Combinatorics, Apr 6, 2012

Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We ... more Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We answer his question by presenting an infinite family of planar hypohamiltonian oriented graphs, the smallest of which has order 9. A computer search showed that 9 is the smallest possible order of a hypohamiltonian oriented graph.

Discrete Mathematics, May 1, 2007

We determine the smallest claw-free, 2-connected, nontraceable graphs and use one of these graphs... more 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.

Discussiones Mathematicae Graph Theory, 1997

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every sub... more 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 P 1 ,. .. , P n be properties of graphs. We say that a graph G has property P 1 • • • • •P n if the vertex set of G can be partitioned into n sets V 1 ,. .. , V n such that the subgraph of G induced by V i has property P i ; i = 1,. .. , n. A hereditary property R is said to be reducible if there exist two hereditary properties P 1 and P 2 such that R = P 1 •P 2. If P is a hereditary property, then a graph G is called P-maximal 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.

Electronic Journal of Combinatorics, Jul 19, 2005

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Ka... more Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ 3n−2 2 −2 for n ≥ 20 and g(n) ≤ 3n−2 2