Inner zonality in graphs. Issue 3 (3rd July 2022)

• Record Type: Journal Article
• Authors: Bowling, Andrew; Zhang, Ping
• Is Part Of: International journal of computer mathematics. Volume 7:Issue 3(2022)
• Page Start: 192
• Page End: 206
• Publication Date: 2022-07-03
• Subjects: Planar graph -- zonal labelling -- inner zonal labelling -- zonal graph -- inner zonal graph; 05C10 -- 05C15 -- 05C78; Computer systems -- Periodicals; Computer systems; Periodicals; 004
• DOI: 10.1080/23799927.2022.2113745
• Languages: English
• ISSNs: 2379-9927

Abstract

Abstract : A zonal labelling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labelling is a zonal graph. A cubic map is a connected 3-regular bridgeless plane graph. It is known that if an independent proof could be given that every cubic map is zonal, then the Four Color Theorem would follow as a corollary. As a step in this direction, it is shown that certain subgraphs of cubic maps are nearly zonal.