Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

Covering of graphs by complete bipartite subgraphs; Complexity of 0–1 matrices

We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log₂ n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n ²/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarján. Dedicated to Paul Erdős on his seventieth birthday     

Solving systems of linear equations over lie nilpotent rings

We show how the so called right and left determinants can be used in the solution of certain systems of linear equations over Lie niipotent rings, A close analogue of Cramer's rule is formulated for right linear equations over the Grassmann algebra.     

Upper bound on the order of τ-critical hypergraphs

The right order of magnitude for the maximal number of vertices in an r-uniform τ-critical hypergraph H is achieved by obtaining an upper bound of O(τ(H)^r?1).     

A characterization of graphs without long induced paths

In a connected graph define the k-center as the set of vertices whose distance from any other vertex is at most k. We say that a vertex set S d-dominates G if for every vertex x there is a y ∈ S whose distance from x is at most d. Call a graph P_t-free if it does not contain a path on t vertices as an induced subgraph. We prove that a connected graph is P_2k-1-free (P_2k-free) if and only if each of its connected induced subgraphs H satisfy the following property: The k-center of H (k - 1)-dominates ((k - 2)-dominates) H. Moreover, we show that the subgraph induced by the (t - 3)-center in any P_t-free connected graph is again connected and has diameter at most t - 3.     

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ(n^3.5logn); but the best combinatorial algorithms have approximation ratio 4/5 only, achieved in O(n²) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O(n²) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.