Alexandr V. Kostochka

Lower bound of the hadwiger number of graphs by their average degree

The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.

On an upper bound of a graph's chromatic number, depending on the graph's degree and density

Abstract Grunbaums conjecture on the existence of k-chromatic graphs of degree k and girth g for every k ≥ 3, g ≥ 3 is disproved. In particular, the bound obtained states that the chromatic number of a triangle-free graph does not exceed [ 3(σ + 2) 4 ], where σ is the graphs degree.

A short proof of the hajnal–szemerédi theorem on equitable colouring

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemeredi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

Note to the paper of Grünbaum on acyclic colorings

In this note counterexamples to some questions of Grunbaum on acyclic colorings are constructed.

On the maximum average degree and the oriented chromatic number of a graph

Abstract The oriented chromatic number o( H ) of an oriented graph H is defined as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′. The oriented chromatic number o( G ) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o( G ) and mad( G ) defined as the maximum average degree of the subgraphs of G.

Choosability conjectures and multicircuits

Abstract This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′= χ ′ for every multigraph, is shown to imply that if a line graph is (a : b) -choosable, then it is (ta : tb) -choosable for every positive integer t. It is proved that ch( H 2 )= χ ( H 2 ) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″( C )= χ ″( C ) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H 2 or T ( C ) is (a : b) -choosable, then it is (ta : tb) -choosable for every positive integer t.

The total coloring of a multigraph with maximal degree 4

Abstract In this paper it is proved that every multigraph with maximal degree 4 has a total coloring in six colors.

Total Colourings of Planar Graphs with Large Girth

It is proved that ifGis a planar graph with total (vertex?edge) chromatic number ??, maximum degree ? and girthg, then ??=?+1 if ??5 andg?5, or ??4 andg?6, or ??3 andg?10. These results hold also for graphs in the projective plane, torus and Klein bottle.

A CLASS OF CONSTRUCTIONS FOR TURAN'S (3, 4)-PROBLEM

Letf(n) denote the minimal number of edges of a 3-uniform hypergraphG=(V, E) onn vertices such that for every quadrupleY ⊂V there existsY ⊃e ∈E. Turán conjectured thatf(3k)=k(k−1)(2k−1). We prove that if Turán’s conjecture is correct then there exist at least 2k−2 non-isomorphic extremal hypergraphs on 3k vertices.

On Sufficient Degree Conditions for a Graph to be

A graph is