查晓亚.Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle[J].数学研究及应用,2019,39(6):581~592 
Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle 
Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle 
投稿时间：20190922 修订日期：20191010 
DOI：10.3770/j.issn:20952651.2019.06.005 
中文关键词: surface planar graph edge partition thickness outerthickness caterpillar tree projective plane Klein bottle 
英文关键词:surface planar graph edge partition thickness outerthickness caterpillar tree projective plane Klein bottle 
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中文摘要: 
In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every nonplanar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every nonplanar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every nonplanar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$. 
英文摘要: 
In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every nonplanar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every nonplanar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every nonplanar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$. 
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