The biparticity of a graph

Abstract

The biparticity P(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, P ( G ) = {10g2x(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.Dedicated to the memory of Paul Turhn BlPARTlClTYThis graphical invariant was defined in [3] along with several other concepts relating to coverings and packings of graphs. The biparticity P(G) is the minimum number of spanning bipartite subgraphs which cover the lines of G. We will determine the biparticity of any graph G in terms of its chromatic number.Our notation and terminology will follow the book [2]. As usual, write {x} for the least integer not less than the real number x. Let E ( G ) be the set of lines of G, V(G) the set of points, and p(@) the number of points. THEOREM. For any graph G, P ( G )Proof. For the upper bound we will proceed by induction to show that if x(G) = 2", then P(G) I n. This is obviously true if n = 1. Assume it is true for n = k, and that x(G) = 2'". Let G, be the subgraph generated by 2' of the colors, and let Gz be the subgraph generated by the other 2' colors. Then x ( Gi) = 2k. By induction p ( G) 5 1 f max P ( G i ) 5 1 + k as required.

On the eulericity of a graph

“…Harary [3] has defined the biparticity P(G) of a connected graph G, as the least number of bipartite subgraphs of G which together cover the lines of G. Harary, HSU, and Miller [4] have shown that if x ( G ) is the chromatic number of G, then…”

Decompositions of Complete Multigraphs Related to Hadamard Matrices

“…In turn, the Singleton bound is a special case of Lemma 6 below with G=+K v . Equality is always attained in Lemma 6 for simple graphs [4].…”

“…Consequently, if a Hadamard matrix of order 4k exists then bp(2kK 4k ) 4k&1. Equality follows from inequality (4 Deleting the first column of the matrix H above and the 2k rows of H with first entry equal to 1, leaves a 2k by 4k&2 matrix B such that every two rows of B have oppositely signed entries in exactly 2k positions. Thus B is an incidence matrix of a decomposition of 2kK 2k into 4k&2 bicliques.…”

Eigensharp graphs: decomposition into complete bipartite subgraphs

“…the minimum number of edge-disjoint subgraphs of G that belong to F and cover the edges of G. The parameter is welldefined as long as K2 G F, where Kn denotes the complete graph or clique on n vertices. The parameter Ty(G) has been studied for several families F: F_I rF(G)_ complete graphs clique partition number forests arboricity paths and cycles (see [6]) all bipartite graphs biparticity (see [4]) matchings edge-chromatic number stars vertex cover number…”

“…Lovasz's motivation in [6] was the long-standing conjecture of Gallai that every n-vertex graph can be decomposed into [n/2] paths; Lovasz proved that [n/2\ paths and/or cycles will suffice. Harary, Hsu, and Miller [4] proved that for every graph G the minimum number of bipartite graphs that partition its edges is exactly [log2 x(G)], where x(G) denotes the ordinary chromatic number. The edge-chromatic number and vertex cover number (minimum number of vertices needed to have an endpoint of each edge) are well-studied parameters with many applications.…”