John Mitchem

A seven-color theorem on the sphere

Ringel has shown that the set of vertices and regions of any normal map on the sphere can be admissibly colored by six colors. In this paper, it is shown that the set of vertices, edges and regions of any normal map on the sphere can be admissibly colored with seven colors.

Star arboricity of graphs

Abstract We develop a connection between vertex coloring in graphs and star arboricity which allows us to prove that every planar graph has star arboricity at most 5. This settles an open problem raised independently by Algor and Alon and by Ringel. We also show that deciding if a graph has star arboricity 2 is NP-complete, even for 2-degenerate graphs.

Bipartite graphs with cycles of all even lengths

Let G = (X, Y, E) be a bipartite graph with X = Y = n. Chvatal gave a condition on the vertex degrees of X and Y which implies that G contains a Hamiltonian cycle. It is proved here that this condition also implies that G contains cycles of every even length when n > 3.

An upper bound for the harmonious chromatic number of a graph

An upper bound for the harmonious chromatic number of a graph G is given. Three corollaries of the theorem are theorems or improvements of the theorems of Miller and Pritikin.

On the harmonious chromatic number of a graph

The harmonious chromatic number of a graph G, denoted by h(G), is the least number of colors which can be assigned to the vertices of G such that adjacent vertices are colored differently and any two distinct edges have different color pairs. This is a slight variation of a definition given independently by Hopcroft and Krishnamoorthy and by Frank, Harary, and Plantholt. D. Johnson has shown that determining h(G) is an NP-complete problem. In this paper we give various other theorems on harmonious chromatic number and discuss various open questions.

An extension of Brooks' theorem to n-degenerate graphs

Abstract For n⪖0,pn(G) denotes the Lick-White vertex-partition number of C In this paper generalized Kempe paths are used to prove that p n (G)⩽{ Δ(G) (n+1)} if G is not an odd cycle, an (n+1)-regular graph, nor a complete graph on t(n+1)+1 vertices. This result generalizes theorems of Brooks and Matula.

On the cost chromatic number of outerplanar, planar, and line graphs

We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost chromatic number of maximal outerplanar and maximal planar graphs can be arbitrarily large and construct several infinite classes of counterexamples to a conjecture of Harary and Plantholt on the cost chromatic number of line graphs.

On the cost-chromatic number of graphs

Abstract We consider vertex colorings in which each color has an associated cost, incurred each time the color is assigned to a vertex. For a given set of costs, a minimum-cost coloring is a vertex coloring which makes the total cost of coloring the graph as small as possible. The cost-chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum-cost coloring of the graph. We establish upper bounds on the cost-chromatic number, show that trees and planar blocks can have arbitrarily large cost-chromatic number, and show that cost-chromatic number is not monotonic with subgraph inclusion.

The entire chromatic number of a normal graph is at most seven

A multigraph is said to be normal if it is embedded in the plane such that each vertex is adjacent to exactly three edges and three regions. In [2], G. Ringel showed that the vertices and regions of a normal multigraph can be colored with six colors such that adjacent elements are colored differently. In this note we consider the problem of coloring vertices, regions, and edges of normal multigraphs. Formally, the entire chromatic number of a plane multigraph G is the fewest number of colors required to color the vertices, regions, and edges of G so that adjacent elements are colored differently. Here a region is adjacent to the vertices and edges which are on its boundary. Also a vertex is adjacent to its incident edges. In [1], H. Izbicki reported that by assuming the four color conjecture M. Neuberger has proved

Uniquelyk-arborable graphs

A graph is uniquelyk-arborable if its point-arboricity isk and there is only one acyclic partition of its point set intok subsets. Several properties of uniquelyk-arborable graphs are presented. One such property is that uniquelyk-arborable graphs are (k−1)-connected. Furthermore, it is shown that for any positive integerk there is a uniquelyk-arborable graph which is notk-connected.