Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2021-03-10 , DOI: 10.1007/s00373-021-02293-x Hongliang Lu , Zixuan Yang , Xuechun Zhang
A graph G has the strong parity property if for every subset \(X\subseteq V(G)\) with |X| even, G has a spanning subgraph F with minimum degree at least one such that \(d_F(v)\equiv 1\pmod 2\) for all \(v\in X\), \(d_F(y)\equiv 0\pmod 2\) for all \(y\in V(G)-X\). Bujtás et al. (Graphs Combin 36(2020):1391–1399, 2020) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujtás, Jendrol’ and Tuza.
中文翻译:
具有强奇偶因子的图的刻画
如果对于每个具有| |的子集\(X \ subseteq V(G)\),则图G具有较强的奇偶性。X | 甚至,G都有一个最小度至少为1的生成子图F,使得所有\(v \ in X \),\(d_F(y)\ equiv 0的\ {d_F(v)\ equiv 1 \ pmod 2 \)\ pmod 2 \)全部\(y \ in V(G)-X \)。Bujtás等。(Graphs Combin 36(2020):1391-1399,2020)引入了这一概念,并推测每个最小程度至少为3的2边连接图都具有很强的奇偶性。在本文中,我们对具有强奇偶性的图进行了刻画,并构造了一个反例来反驳Bujtás,Jendrol'和Tuza提出的猜想。