We study the (1 : b) Maker–Breaker component game, played on the edge set of a d-regular graph. MAKER’s aim in this game is to build a large connected component, while BREAKER’s aim is to not let him do so. For all values of BREAKER’s bias b, we determine whether BREAKER wins (on any d-regular graph) or MAKER wins (on almost every d-regular graph) and provide explicit winning strategies for bot...

We consider the bandwidth coloring problem, a generalization of the well-known graph coloring problem. For the latter problem, a classical theorem, discovered independently by Gallai, Roy and Vitaver, states that the chromatic number of a graph is bounded from above by the number of vertices in the longest elementary path in any directed graph derived by orienting all edges in the graph. We gen...

We prove that the threshold for a random graph to have a k-core is equal to the threshold for having a subgraph which meets a necessary condition of Gallai for being k-critical. 1991 Mathematics Subject Classification: Primary 05C80; Secondary 05C15.

Gallai-colorings of complete graphs – edge colorings such that no triangle is colored with three distinct colors – occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result – any Gallai-colored complete gr...

Gallai and Roy proved that a graph is k-colorable if and only if it has an orientation without directed paths of length k. We initiate the study of analogous characterizations for the existence of generalized graph colorings, where each color class induces a subgraph satisfying a given (hereditary) property. It is shown that a graph is partitionable into at most k independent sets and one induc...

The path number p(G) of a graph G is the minimum number of paths needed to partition the edge set of G. Gallai conjectured that p(G) b n+1 2 c for every connected graph G of order n. Because of the graph consisting of disjoint triangles, the best one could hope for in the disconnected case is p(G) b 2 3 nc. We prove the sharper result that p(G) 1 2 u + b 2 3 gc where u is the number of odd vert...

A graph G is k-critical if every proper subgraph of G is (k − 1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100 logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k − 1) logn/ log(k − 2). We thus settle the probl...

In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...

A star-factor of a graph G is a spanning subgraph of G such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph G is called star-uniform if all star-factors of G have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree an...