Stefan Grünewald

Harmonic graphs with small number of cycles

Let G be a graph on n vertices v1,v2,...,vn and let d(vi) be the degree (= number of first neighbors) of the vertex vi. If (d(v1),d(v2),...,d(vn))^t is an eigenvector of the (0,1)-adjacency matrix of G, then G is said to be harmonic. Earlier all harmonic trees were determined; their number is infinite. We now show that for any c,c>1, the number of connected harmonic graphs with cyclomatic number c is finite. In particular, there are no connected non-regular unicyclic and bicyclic harmonic graphs and there exist exactly four and eighteen connected non-regular tricyclic and tetracyclic harmonic graphs.

Chromatic-index-critical graphs of orders 13 and 14

A graph is chromatic-index-critical if it cannot be edge-coloured with @D colours (with @D the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order [[18] A.J.W. Hilton, R.J. Wilson, Edge-colorings of graphs: a progress report, in: M.F. Cabobianco, et al. (Eds.), Graph Theory and its Applications: East and West, New York, 1989, pp. 241-249; [31] H.P. Yap, Some topics in graph theory, London Mathematical Society, Lecture Note Series, vol. 108, Cambridge University Press, Cambridge, 1986]. In this paper we show that there are no chromatic-index-critical graphs of order 14. Our result extends that of [[5] G. Brinkmann, E. Steffen, Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19 (1998) 889-900] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexamples given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13.

Bounds for the Independence Number of Critical Graphs

In 1968 Vizing conjectured that any independent vertex set of an edge-chromatic critical graph G contains at most half of the vertices of G , that is, α( G ) [les ] ½[mid ] V ( G )[mid ]. Let Δ be the maximum vertex degree in a critical graph. For each Δ, we determine c (Δ) such that α( G ) [les ] c (Δ)[mid ] V )[mid ].

Integer linear programming as a tool for constructing trees from quartet data

The task of the quartet puzzling problem is to find a best-fitting binary X-tree for a finite n-set from confidence values for the 3n4 binary trees with exactly four leaves from X, its fitness being measured by the sum of the confidence values of all induced four-leaves subtrees. We describe a method for finding an exact solution of this problem by integer linear programming. Similar procedures can also be used for finding, e.g. best-fitting circular networks. A crucial problem in this context is, of course, how to obtain the input confidence values for the quartet trees. We propose to use inner products of rate-matrix diagonals calculated for pairs of taxa and present the trees resulting from applying our approach to two data sets of up to 36 mitochondrial sequences of mammals including an outgroup.

Independent sets and 2-factors in edge-chromatic-critical graphs: EDGE-CHROMATIC-CRITICAL GRAPHS

In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2-factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those graphs.

Chromatic-index-critical graphs of even order

We shall prove that for any graph H that is a core, if χ(G) is large enough, then H × G is uniquely H-colorable. We also give a new construction of triangle free graphs, which are uniquely n-colorable.

Semiharmonic trees and monocyclic graphs

Abstract A graph G is defined to be semiharmonic if there is a constant μ (necessarily a natural number) such that, for every vertex v , the number of walks of length 3 starting in v equals μd G ( v ) where d G ( v ) is the degree of v . We determine all finite semiharmonic trees and monocyclic graphs.

Semiharmonic graphs with fixed cyclomatic number

Abstract Let the trunk of a graph G be the graph obtained by removing all leaves of G. We prove that, for every integer c ≥ 2, there are at most finitely many trunks of semiharmonic graphs with cyclomatic number c—in contrast to the (act established by the last two of the present authors in their paper Semiharmonic Bicyclic Graphs (this journal) that there are infinitely many connected semiharmonic graphs with given cyclomatic number. Further, we prove that there are at most finitely many semiharmonic but not almost semiregular graphs with cyclomatic number c.

Semiharmonic bicyclic graphs

Classification of harmonic and semiharmonic graphs according to their cyclomatic number became of interest recently. All finite harmonic graphs with up to four independent cycles, as well as all finite semiharmonic graphs with at most one cycle were determined. Here, we determine all finite semiharmonic bicyclic graphs. Besides that, we present several methods for constructing semiharmonic graphs from existing ones, and we apply one of these constructions to show that the number of semiharmonic graphs with fixed cyclomatic number k is infinite for every k.

Cyclically 5-edge connected non-bicritical critical snarks

Snarks are bridgeless cubic graphs with chromatic index χ = 4. A snark G is called critical if χ(G − {v, w}) = 3, for any two adjacent vertices v and w. For any k ≥ 2 we construct cyclically 5-edge connected critical snarks G having an independent set I of at least k vertices such that χ(G − I) = 4. For k = 2 this solves a problem of Nedela and Škoviera [6].