-
D. R. Woodall
[show abstract]
[hide abstract]
ABSTRACT: The concept of subcontraction-equivalence is defined, and 14 graph-theoretic properties are exhibited that are all subcontraction-equivalent if Hadwiger's conjecture is true. Some subsets of these properties are proved to be subcontraction-equivalent anyway. Hadwiger's conjecture is expressed as the union of three independent and strictly weaker subconjectures. As a first step toward one of these subconjectures, it is proved that a graph that does not have Km+1 as a subcontraction must contain an independent set consisting of at least 1/2(m − 1) of its vertices.
Journal of Graph Theory 10/2006; 11(2):197 - 204. · 0.52 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: Let X, T and C be, respectively, a finite set with at least three points, a set of ordered triples of distinct points from X, and a cyclic ordering of the points in X. Define T
Í \subseteq
C to mean that, for every a b c Í \subseteq
C. We say that T is noncyclic if C(T) is empty, cyclic if C(T) is nonempty, uniquely cyclic if | C(T) | = 1, a partial cycle order if it is cyclic and T ={a b c :{a b c}
Í \subseteq
C for all C C(T)}, and a total cycle order if it is a uniquely cyclic partial cycle order. Many years ago E. V. Huntington axiomatized total cycle orders by independent necessary and sufficient conditions on T. The present paper studies the more relaxed structures of cyclic T sets and partial cycle orders. We focus on conditions for cyclicity, a theory of cycle dimension of partial cycle orders, and extremal problems that address combinatorial structures of T sets.
Order 05/1999; 16(2):149-164. · 0.41 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: It is proved that a planar graph with maximum degree Δ ≥ 11 has total (vertex-edge) chromatic number $Delta; + 1. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 53–59, 1997
Journal of Graph Theory 12/1998; 26(1):53 - 59. · 0.52 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified
maximum vertex degree.
Mathematical Notes 01/1998; 64(5):562-570. · 0.29 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: We exploit the technique of Galvin (1995) to prove that an orientation D of a line-graph G (of a multigraph) is kernel-perfect if and only if every oriented odd cycle in D has a chord (or pseudochord) and every clique has a kernel.
Discrete Mathematics.
-
[show abstract]
[hide abstract]
ABSTRACT: This paper and its successor(s) aim to derive a mathematical description of bond graphs in general and of their junction structures in particular. It also introduces bond graphs to mathematicians that have no previous knowledge of them. In this introductory paper, a definition of bond graphs is given and the concept of acausal equivalence is introduced. Fifteen basic operations are defined and proved to be acausal equivalence operations. It is proved that these basic operations form a complete set, in the sense that, if two bond graphs are acausally equivalent, then each can be converted into the other by a sequence of these operations and their inverses. In the course of the proof it is shown that every bond graph is acausally equivalent to one in a standard form. These standard bond graphs are used to demonstrate various mathematical properties of bond graphs, and to derive a new procedure for testing whether or not a given set of input variables uniquely determines the corresponding set of output variables. This should be of interest to mathematicians and engineers alike.
Discrete Applied Mathematics.
-
[show abstract]
[hide abstract]
ABSTRACT: Electrical networks are defined and a definition of when a bond graph and an electrical network are equivalent is given. Bond graphs and electrical networks are defined to be primitive if they contain no transformers or gyrators. A bond graph is defined to be realisable if it is equivalent to an electrical network and primitively realisable if it is equivalent to a primitive electrical network. It is shown how to construct a bond graph equivalent to a given electrical network and how to construct an electrical network equivalent to a given bond graph. Chordless odd loops are defined and a characterisation of primitively realisable bond graphs in terms of chordless odd loops and forbidden induced subgraphs is given. It is shown how to construct a primitive network equivalent to a given primitively realisable bond graph.
Discrete Applied Mathematics.
-
[show abstract]
[hide abstract]
ABSTRACT: This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B63(1995), 153–158), who proved that the list edge chromatic numberχ′list(G) of a bipartite multigraphGequals its edge chromatic numberχ′(G). It is now proved here that if every edgee=uwof a bipartite multigraphGis assigned a list of at least max{d(u), d(w)} colours, thenGcan be edge-coloured with each edge receiving a colour from its list. If every edgee=uwin an arbitrary multigraphGis assigned a list of at least max{d(u), d(w)}+⌊min{d(u), d(w)}⌋ colours, then the same holds; in particular, ifGhas maximum degreeΔ=Δ(G) thenχ′list(G)⩽⌊Δ⌋. Sufficient conditions are given in terms of the maximum degree and maximum average degree ofGin order thatχ′list(G)=Δandχ″list(G)=Δ+1. Consequences are deduced for planar graphs in terms of their maximum degree and girth, and it is also proved that ifGis a simple planar graph andΔ⩾12 thenχ′list(G)=Δandχ″list(G)=Δ+1.
Journal of Combinatorial Theory, Series B.
