A total-chromatic number analogue of plantholt's theorem

AbstractThe total chromatic number XT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no two adjacent vertices receive the same colour, no two edges incident with the same vertex receive the same colour, and no edge receives the same colour as either of the vertices it is incident with.Let n⩾1, let J be a subgraph of K2n, let e=∥E(J)∥ and let j(J) be the maximum size of a matching in J. Then XT(K2n[bsol]E(J))=2n+1 if and only if e+j≤n−1

The total chromatic number of nearly complete bipartite graphs

AbstractThe total chromatic number χT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no two adjacent vertices receive the same colour, no two edges incident with the same vertex receive the same colour, and no edge receives the same colour as either of the vertices it is incident with. Let n ≥ 1, let J be a subgraph of Kn,n, let e = |E(J)|, and let j(J) be the maximum size (i.e., number of edges) of a matching in J. Then χT(Kn,nE(J)) = n + 2 if and only if e + j ≤ n − 1

A Fast Algorithm for Computing a Nearly Equitable Edge Coloring with Balanced Conditions

We discuss the nearly equitable edge coloring problem on a multigraph and propose an ecient algorithm for solving the problem, which has a better time complexity than the previous algorithms. The coloring computed by our algorithm satises additional balanced conditions on the number of edges used in each color class, where conditions are imposed on the balance among all edges in the multigraph as well as the balance among parallel edges between each vertex pair. None of the previous algorithms are guaranteed to satisfy these balanced conditions simultaneously. To achieve these improvements, we propose a new recoloring procedure, which is based on a set of edge-disjoint alternating walks, while the existing algorithms are based on an Eulerian circuit or a single alternating walk. This new recoloring procedure makes it possible to reduce the time complexity of the algorithm.

Cycle decompositions of the complete graph

For a positive integer n, let G be K n if n is odd and K n less a one-factor if n is even. In this paper it is shown that, for non-negative integers p, q and r, there is a decomposition of G into p 4-cycles, q 6-cycles and r 8-cycles if 4p + 6q + 8r = |E(G)|, q = 0 if n < 6 and r = 0 if n < 8

Amalgamations of factorizations of complete equipartite graphs,

Let t be a positive integer, and let L=(l1,…,lt) and K=(k1,…,kt) be collections of nonnegative integers. A graph has a (t,K,L) factorization if it can be represented as the edge-disjoint union of factors F1,…,Ft where, for 1it, Fi is ki-regular and at least li-edge-connected. In this paper we consider (t,K,L)-factorizations of complete equipartite graphs. First we show precisely when they exist. Then we solve two embedding problems: we show when a factorization of a complete σ-partite graph can be embedded in a (t,K,L)-factorization of a complete s-partite graph,

An algorithm for finding factorizations of complete graphs,

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge-connectivity of each factor is given