Michael Stiebitz

Decomposing graphs under degree constraints

We prove a conjecture of C. Thomassen: If s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and t, respectively.

A solution to a colouring problem of P. Erdos

A consequence of Theorem 1.1 is that G has independence number n, as conjectured by D.Z. Du and D.F. Hsu in 1986 at MIT. In April 1987 Erdiis visited MIT and took an interest in the conjecture. He formulated the above colouring extension, which soon became known as the ‘cycle plus triangles’problem. It was mentioned in [3-5,8]. Fellows [5] observed that Theorem 1.1 is in fact equivalent to an old conjecture of Schur, that for any partition of the integers Z into triples, there is another partition S1, S,, S, of Z such that each Si contains a member of each triple, but no consecutive pair of integers. The proof of Theorem 1.1 given in Section 2 uses a recent result of Alon and Tarsi [l], see Theorem 1.2 in Subsection 1.2. Alon and Tarsi’s results provides a colouring criteria for a graph G based on orientations of G.

Dirac's map-color theorem for choosability

Let G be an n-vertex graph with list-chromatic number Cl. Suppose that each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least tn-Cl vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a corollary, we show that at least

On a special case of Hadwiger's conjecture

{6}\over{7}

K5 is the only double-critical 5-chromatic graph

of the conjectured number can be colored.

Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs

Hadwigers Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case.

ON THE NUMBER OF EDGES IN COLOUR-CRITICAL GRAPHS AND HYPERGRAPHS

Abstract A connected graph G is called double-critical if the chromatic number of G decreases by two if any two adjacent vertices of G are removed. L. Lovasz conjectured that Kk is the only double-critical graph with chromatic number k. This is almost trivial for k⩽4 and the aim of this note is to prove this conjecture for k = 5.

Subgraphs of colour-critical graphs

Tibor Gallai made the following conjecture. LetG be ak-chromatic colour-critical graph. LetL denote the set of those vertices ofG having valencyk−1 and letH be the rest ofV(G). Then the number of components induced byL is not less than the number of components induced byH, providedL ≠ 0.We prove this conjecture in a slightly generalized form.

A new lower bound on the number of edges in colour-critical graphs and hypergraphs

G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.

On constructive methods in the theory of colour-critical graphs

Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed.In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastckn2 edges must be removed.In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.