Bing Zhou

The star chromatic number of a graph

We study a generalization of the notion of the chromatic number of a graph in which the colors assigned to adjacent vertices are required to be, in a certain sense, far apart.

Bounded vertex coloring of trees

Abstract A k -bounded vertex coloring of a graph G is a usual vertex coloring in which each color is applied to at most k vertices. The bounded chromatic number χ k (G) is the smallest number of colors such that G admits a k -bounded coloring. In this paper, we study the problem of bounded vertex coloring of trees. We characterize the trees on n vertices that can be k -bounded colored with ⌈n/k⌉ colors. The analysis provides an efficient algorithm for determining the k -bounded chromatic number of a tree and an optimal coloring. This answers a question raised by Hansen, Hertz and Kuplinsky (Discrete Math. 111 (1993), 305).

The edge density of 4-critical planar graphs

Several constructions of 4-critical planar graphs are given. These provide answers to two questions of B. Grünbaum and give improved bounds for the maximum edge density of such graphs.

Representing an ordered set as the intersection of super greedy linear extensions

A linear extension [x1<x2<...<xt] of a finite ordered set P=(P, <) is super greedy if it can be obtained using the following procedure: Choose x1 to be a minimal element of P; suppose x1,...,xi have been chosen; define p(x) to be the largest j≤i such that xj<x if such a j exists and 0 otherwise; choose xi+1 to be a minimal element of P-{x1,...,xi} which maximizes p. Every finite ordered set P can be represented as the intersection of a family of super greedy linear extensions, called a super greedy realizer of P. The super greedy dimension of P is the minimum cardinality of a super greedy realizer of P. Best possible upper bounds for the super greedy dimension of P are derived in terms of |P-A| and width (P-A), where A is a maximal antichain.

On a construction of graphs with high chromatic capacity and large girth

The chromatic capacity of a graph G,@gCAP(G), is the largest integer k such that there is a k-colouring of the edges of G such that when the vertices of G are coloured with the same set of colours, there are always two adjacent vertices that are coloured with the same colour as that of the edge connecting them. It is easy to see that @gCAP(G)@[emailxa0protected](G)-1. In this note we present a construction based on the idea of classic construction due to B. Descartes for graphs G such that @gCAP(G)[emailxa0protected](G)-1 and G does not contain any cycles of length less than q for any given integer q.

Color-critical graphs and hypergraphs with few edges and no short cycles

Abstract We give constructions of color-critical graphs and hypergraphs with no cycles of length 5 or shorter and with relatively few edges.

On a conjecture of Gallai concerning complete subgraphs of k -critical graphs

A graph G is said to be k-critical if it has chromatic number k but every proper subgraph of G has a (k−1)-coloring. T. Gallai asked whether each k-critical graph of order n contains at most n complete subgraphs of order k − 1. This is clearly so when k = n, and it is also true when k = 3 since the only 3-critical graphs are the cycles of odd length. M. Stiebitz recently gave a positive answer to Gallais question in the case k = 4. In this paper we give an affirmative answer for all k⩾5.

A lower bound for the chromatic capacity in terms of the chromatic number of a graph

Abstract When the vertices and edges are coloured with k colours, an edge is called monochromatic if the edge and the two vertices incident with it all have the same colour. The chromatic capacity of a graph G , χ C A P ( G ) , is the largest integer k such that the edges of G can be coloured with k colours in such a way that when the vertices of G are coloured with the same set of colours, there is always a monochromatic edge. It is easy to see that χ C A P ( G ) ≤ χ ( G ) − 1 . Greene has conjectured that there is an unbounded function f such that χ C A P ( G ) ≥ f ( χ ( G ) ) . In this article we prove Greene’s conjecture.

The maximum number of cycles in the complement of a tree

Abstract In this paper we investigate trees on a fixed set of vertices whose complements contain the maximum possible number of cycles. Let T be a tree and c ( T ′) be the number of cycles in the complement of T . We prove that for every tree T of n ⩾6 vertices and with diameter d between 4 and n −2 inclusive, there is a tree T 1 of n vertices with diameter at least d +1 so that c ( T ′ 1 )> c ( T ′). We further deduce that among all trees of n ⩾9 vertices, a path on n vertices has the maximum number of cycles in its complement. This settles in the affirmative a conjecture of K.B. Reid.

LARGE FACES IN 4-CRITICAL PLANAR GRAPHS WITH MINIMUM DEGREE 4

AbstractWe prove that the size of the largest face of a 4-critical planar graph with δ≥4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that showsn