Gao, Y., Farahani, M., Gao, W. (2017). A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs. Transactions on Combinatorics, 6(1), 13-19. doi: 10.22108/toc.2017.20355

Yun Gao; Mohammad Reza Farahani; Wei Gao. "A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs". Transactions on Combinatorics, 6, 1, 2017, 13-19. doi: 10.22108/toc.2017.20355

Gao, Y., Farahani, M., Gao, W. (2017). 'A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs', Transactions on Combinatorics, 6(1), pp. 13-19. doi: 10.22108/toc.2017.20355

Gao, Y., Farahani, M., Gao, W. A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs. Transactions on Combinatorics, 2017; 6(1): 13-19. doi: 10.22108/toc.2017.20355

A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs

^{1}Department of Editorial, Yunnan Normal University

^{2}Department of Applied Mathematics, Iran University of Science and Technology

^{3}School of Information and Technology, Yunnan Normal University

Abstract

A graph $G$ is called a fractional $(k,n',m)$-critical deleted graph if any $n'$ vertices are removed from $G$ the resulting graph is a fractional $(k,m)$-deleted graph. In this paper, we prove that for integers $k\ge 2$, $n',m\ge0$, $n\ge8k+n'+4m-7$, and $\delta(G)\ge k+n'+m$, if $$|N_{G}(x)\cup N_{G}(y)|\ge\frac{n+n'}{2}$$ for each pair of non-adjacent vertices $x$, $y$ of $G$, then $G$ is a fractional $(k,n',m)$-critical deleted graph. The bounds for neighborhood union condition, the order $n$ and the minimum degree $\delta(G)$ of $G$ are all sharp.

[1] J. A. Bondy and U. S. R. Mutry, Graph theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.

[2] W. Gao, Some results on fractional deleted graphs, Do ctoral disdertation of So o chow university, 2012.

[3] W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientic World Jo., 2014, Article ID 369798, PP. 7, http://dx.doi.org/10.1155/2014/369798.

[4] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014) 55-65.

[5] W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Util. Math., 98 (2015) 295-310.

[6] G. Liu and L. Zhang, Toughness and the existence of fractional k -factors of graphs, Discrete Math., 308 (2008) 1741-1748.

[7] J. Yu, G. Liu, M. Ma and B. Cao, A degree condition for graphs to have fractional factors, Adv. Math. (China), 35 (2006) 621-628.

[8] S. Z. Zhou, A minimum degree condition of fractional ( k ; m )-deleted graphs, C. R. Math. Acad. Sci., Paris, 347 (2009) 1223-1226.

[9] S. Z. Zhou, A neighb orho o d condition for graphs to b e fractional (k,m)-deleted graphs, Glasg. Math. J., 52 (2010) 33-40.

[10] S. Z. Zhou, A suﬃcient condition for a graph to b e a fractional ( f ; n )-critical graph, Glasg. Math. J. , 52 (2010) 409-415.

[11] S. Z. Zhou, A suﬃcient condition for graphs to b e fractional ( k ; m )-deleted graphs, Appl. Math. Lett., 24 (2011) 1533-1538.

[12] S. Z. Zhou and Q. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015) 125-133.