Eckhard Steffen

Classifications and characterizations of snarks

Abstract In this paper vertex-reductions of snarks are considered. Each reduction naturally divides the class of snarks into three classes. The first class contains those snarks which are not reducible to a snark. The second contains those which are not reducible to a 3-colorable graph and the third one consists of the snarks which are reducible to a snark as well as to a 3-colorable graph. We distinguish three kinds of vertex-reductions according to the distance between the vertices used for the reduction. We give several characterizations of the induced classes, and we give structural characterizations in terms of 2-factors. We construct infinite families of snarks belonging to a certain class and analyze the relations between the classes.

Measurements of edge-uncolorability

Abstract Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k =2,3, let c k be the maximum size of a k -colorable subgraph of a cubic graph G =( V , E ). We consider r 3 =| E |− c 3 and r 2 = 2 3 |E|−c 2 . We show that on one side r 3 and r 2 bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G , the smallest possible number of odd circuits in a 2-factor of G . We construct cyclically 5-edge connected cubic graphs where r 3 and ω are arbitrarily far apart, and show that for each 1⩽ c ω ⩾ cr 3 . For k =2,3, let ζ k denote the largest fraction of edges that can be k -colored. We give best possible bounds for these parameters, and relate them to each other.

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number χ(G) and the chromatic number χ(G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each ε>0 and eachm≥2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+ε. This answers another question asked by Abbott and Zhou.

Chromatic-index-critical graphs of orders 11 and 12

Abstract A chromatic-index-critical graphGonnvertices is non-trivial if it has at most Δ ⌊ n 2 ⌋ edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic-index-critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromatic-index-critical graphs of order ≤12.

1-Factor and Cycle Covers of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let Ei be the set of edges contained in precisely i members of the k 1-factors. Let µkG be the smallest |E0| over all lists of k 1-factors of G. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if µ3Gi¾?0, then 2µ3G is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with µ4G=0 have a 4-cycle cover of length 43|EG| and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang . We also give a negative answer to a problem stated in .

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 ].

3- and 4-critical graphs of small even order

We show that there is exactly one 3-critical graph of order 22 and that there are exactly nine of order 24. The graph of order 22 is shown to be the smallest 3-critical graph of even order. We also show that there are exactly two 4-critical graphs of order 18, and that these graphs are the smallest 4-critical graphs of even order. The results are obtained with the aid of a computer.

Choosability in signed planar graphs

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ? { 3 , 4 , 5 , 6 } , every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.

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.

Reduction of symmetric configurations n 3

Abstract Symmetric configurations n 3 are equivalent to the bicubic graphs of girth ⩾6. They are reducible to the configuration 7 3 , the Fano–Heawood-graph F , i.e. the projective plane of order 2 by means of a Martinetti-like procedure.