Papers by Ortrud Oellermann
The Electronic Journal of Combinatorics, 2011
Let $G$ be a graph. A Hamilton path in $G$ is a path containing every vertex of $G$. The graph $G... more Let $G$ be a graph. A Hamilton path in $G$ is a path containing every vertex of $G$. The graph $G$ is traceable if it contains a Hamilton path, while $G$ is $k$-traceable if every induced subgraph of $G$ of order $k$ is traceable. In this paper, we study hamiltonicity of $k$-traceable graphs. For $k \geq 2$ an integer, we define $H(k)$ to be the largest integer such that there exists a $k$-traceable graph of order $H(k)$ that is nonhamiltonian. For $k \le 10$, we determine the exact value of $H(k)$. For $k \ge 11$, we show that $k+2 \le H(k) \le \frac{1}{2}(3k-5)$.
Graph Theory, 2018
Graph Theory is believed to have begun with the famous Konigsberg Bridge Problem. Figure 1 shows ... more Graph Theory is believed to have begun with the famous Konigsberg Bridge Problem. Figure 1 shows a map of Konigsberg as it appeared in the eighteenth century. The town was spanned by seven bridges passing over the river Pregel and connecting the four land masses on which Konigsberg was built. The townsfolk amused themselves by taking walks through Konigsberg, attempting to cross each of the seven bridges exactly once. In 1736 Euler put an end to their speculation that such a walk did not exist by giving a rigorous argument which proved their conjecture. Motivated by this problem, a graph is called eulerian if it has a closed walk that traverses each edge exactly once. It is well known that a connected graph is eulerian if and only if the degree of each vertex is even. The eulerian problem for connected graphs is thus easily solved by checking whether the degree of each vertex is even. The vertex analogue of eulerian graphs are the Hamiltonian graphs. These are graphs that have a cycle that passes through each vertex exactly once and are named after Sir William Rowan Hamilton who devised the Icosian Game for two players. One of the problems in the game required the first player to select a path of five vertices on the dodecahedron. The second player had to extend this path to a cycle that contained all 20 points of the dodecahedron. If a graph has a cycle that contains all its vertices such a cycle is called a Hamiltonian cycle. In contrast to the eulerian problem, the Hamilton cycle problem, i.e., the problem of determining whether a given graph has a Hamiltonian cycle, has no known simple solution. As a result many sufficient conditions for hamiltonicity have been established. In this chapter we will describe problems and conjectures that have their roots in the Hamilton cycle problem. Open image in new window Fig. 1 A map of the Konigsberg bridges
Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only... more Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V, then the elements of M are called convex sets and the pair (V,M) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ M. Then x ∈ X is an extreme point for X if X \ {x} ∈ M. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V (T) \U is a cut-vertex of the subgraph induced by V (T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural ...
Discrete Mathematics, 2007
A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a... more A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I (S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S = {u, v}, then I (S) = I [u, v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that u,v∈S I [u, v] = V (G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I (S) = V (G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G) sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.
Discrete Mathematics, 1996
Sumner and Blitch defined a graph G to be k-y-critical if 7(G) = k and 7(G + uv) = k -1 for each ... more Sumner and Blitch defined a graph G to be k-y-critical if 7(G) = k and 7(G + uv) = k -1 for each pair u, v of nonadjacent vertices of G. We define a graph to be k-(7,d)-critical if 7(G) = k and 7(G + uv) = k -I for each pair u, v of nonadjacent vertices of G that are at distance at most d apart. The 2-(7, 2)-critical graphs are characterized. Sharp upper bounds on the diameter of 3-(7, 2)-and 4-(7, 2)-critical graphs are established and partial characterizations of 3-(7, 2)-critical graphs are obtained.
Discrete Mathematics, 2006
Discrete Mathematics, 2009
Discrete Mathematics, 2002
Let G1; G2; : : : ; Gt be an arbitrary t-edge colouring of Kn, where for each i ∈ {1; 2; : : : ; ... more Let G1; G2; : : : ; Gt be an arbitrary t-edge colouring of Kn, where for each i ∈ {1; 2; : : : ; t}, Gi is the spanning subgraph of Kn consisting of all edges coloured with colour i. The upper domination Ramsey number u(n1; n2; : : : ; nt) is deÿned as the smallest n such that for every t-edge colouring G1; G2; : : : ; Gt of Kn, there is at least one i ∈ {1; 2; : : : ; t} for which Gi has upper domination number at least ni. We show that 136u(3; 3; 3)614.
Discrete Mathematics, 2011
A family C of sets has the Helly property if any subfamily C ′ whose elements are pairwise inters... more A family C of sets has the Helly property if any subfamily C ′ whose elements are pairwise intersecting has non-empty intersection. Suppose that C is a non-empty family of subsets of a finite set V : the Helly number h(C) of C is the least positive integer n such that every n-wise intersecting subfamily of C has non-empty intersection.
Discrete Mathematics, 2010
A digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable.
Discrete Mathematics, 1996
For any graph G and a set ~ of graphs, two distinct vertices of G are said to be ~adjacent if the... more For any graph G and a set ~ of graphs, two distinct vertices of G are said to be ~adjacent if they are contained in a subgraph of G which is isomorphic to a member of ~. A set S of vertices of G is an ~-dominating set (total ~¢~-dominating set) of G if every vertex in V(G)-S (V(G), respectively) is 9¢g-adjacent to a vertex in S. An ~-dominating set of G in which no two vertices are oCf-adjacent in G is an ~,~-independent dominating set of G. The minimum cardinality of an ~-dominating set, total ~-dominating set and ~-independent dominating set of G is known as the ~-domination number, total ~-domination number, and ~Yf-independent dominating number, of G, denoted, respectively, by 7ae(G),7~(G), and i~e(G).
Discrete Mathematics, 2002
In this paper, we consider the concept of the average connectivity of a graph, deÿning it to be t... more In this paper, we consider the concept of the average connectivity of a graph, deÿning it to be the average, over all pairs of vertices, of the maximum number of internally disjoint paths connecting these vertices. Our main results are sharp bounds on the value of this parameter, and a construction of graphs whose average connectivity is the same as the connectivity. Along the way, we establish some new results on connectivity. : S 0 0 1 2 -3 6 5 X ( 0 1 ) 0 0 1 8 0 -7 p i=1 (p − 2i + 1)d i . Note that for j = p − i + 1 and i 6 p=2 ; (p − 2i + 1)d i + (p − 2j + 1)d j ¿ 0 since d i ¿ d j . Consequently, D ¿ 0. Moreover, D attains its minimum when the degrees are as nearly equal as possible. If d = 2q=p (so r = 2q−pd), this occurs when d 1 = · · · = d r = d+1
Discrete Mathematics, 1986
The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum... more The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum number of vertices required to add to G to construct a d-regular bipartite supergraph of G. It is shown that if G is a connected m-by-n bipartite graph with m <~ n and n -m >~ d -1, then br(G) = n -m. If, however, n -m ~< d -2, then br(G) = n -m + 2/for some l satisfying 0 ~< l ~< d-(n-m). Conversely, if l, k and d (>2) are integers such that 0 <~ l <~ k and 2 <~ k <~ d, then there is an connected m-by-n bipartite graph G of maximum degree d for which br(G) = n -m + 2/, for some m and n with k = d -(n -m).
Discrete Mathematics, 2005
The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an ... more The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighbourhoods are complete graphs. It is known that the class of graphs with the Minkowski-Krein-Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbour is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set.
Discrete Mathematics, 2004
Let G and H be graphs. A graph with colored edges is said to be monochromatic if all its edges ha... more Let G and H be graphs. A graph with colored edges is said to be monochromatic if all its edges have the same color and rainbow if no two of its edges have the same color. Given two bipartite graphs G1 and G2, the bipartite rainbow ramsey number BRR(G1; G2) is the smallest integer N such that any coloring of the edges of KN;N with any number of colors contains a monochromatic copy of G1 or a rainbow copy of G2. It is shown that BRR(G1; G2) exists if and only if G1 is a star or G2 is a star forest. Exact values and bounds for BRR(G1; G2) for various pairs of graphs G1 and G2 for which the bipartite ramsey number is deÿned are established.
Discrete Applied Mathematics, 2000
For an ordered subset W = {w1; w2; : : : ; w k } of vertices in a connected graph G and a vertex ... more For an ordered subset W = {w1; w2; : : : ; w k } of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v | W ) = (d(v; w1), d(v; w2); : : :
Discrete Applied Mathematics, 1996
Let S be a nonempty set of vertices of a connected graph G. Then the Steiner distance of S is the... more Let S be a nonempty set of vertices of a connected graph G. Then the Steiner distance of S is the minimum size (number of edges) of a connected subgraph of G containing S. Let n 3 2 be an integer and suppose that G has at least n vertices. Then the Steiner n-distance of a vertex u of G is defined to be the sum of the Steiner distances of all sets of n vertices that include C. The Steiner n-median M,(G) of G is the subgraph induced by the vertices of minimum Steiner n-distance. We show that the Steiner n-median of a tree is connected and determine those trees that are Steiner n-medians of trees, and show that if T is a tree with more than n vertices, then M,( T ) c M, + 1 (T ). Further, a 0( 1 V( T ) I) algorithm for finding the Steiner n-median of a tree T is presented and a 0( n IV(T)I") algorithm for finding the Steiner n-distances of all vertices in tree T is described. For a connected graph G of order p > n, the n-median value of G is the least Steiner n-distance of any of its vertices. For p > 2n -1, sharp upper and lower bounds for the n-median values of trees of order p are given, and it is shown that among all trees of a given order, the path has maximum n-median value.
Discrete Applied Mathematics, 2007
Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u, v of vertices of... more Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u, v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.
Discrete Applied Mathematics, 2003
In this paper, we consider the concept of the average connectivity of a graph, deÿned to be the a... more In this paper, we consider the concept of the average connectivity of a graph, deÿned to be the average, over all pairs of vertices, of the maximum number of internally disjoint paths connecting these vertices. We establish sharp bounds for this parameter in terms of the average degree and improve one of these bounds for bipartite graphs with perfect matchings. Sharp upper bounds for planar and outerplanar graphs and cartesian products of graphs are established. Nordhaus-Gaddum-type results for this parameter and relationships between the clique number and chromatic number of a graph are also established. ?
Discrete Applied Mathematics, 1998
Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all thos... more Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all those vertices that lie on some shortest u -1; path. Let S be a set of vertices in a connected graph G. Then the Steiner distance d,(S) of S in G is the smallest number of edges in a connected subgraph of G that contains S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval I,(S) of S consists of all those vertices that lie on some Steiner tree for S. Let S be an n-set of vertices of G and suppose that k < n. Then the k-intersection interval of S. denoted by I,(S) is the intersection of all Steiner intervals of all k-subsets of S. It is shown that if s = ;a,, u2, . ..) u,j \ is a set of n 3 2 vertices of a graph G and if the 2-intersection interval of S is nonempty and x~l,(S), then d(S) = C:'= 1 d (u,,x). It is observed that the only graphs for which the ?-intersection intervals of all n-sets, n > 4, are nonempty are stars. Moreover. for every II > 4, those graphs with the property that the 3-intersection interval of every n-set is nonempty are completely characterized.
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Papers by Ortrud Oellermann