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Ruthie Abeliovich and Edwin Seroussi

Open access

Yongxin Lan, Zhongmei Qin and Yongtang Shi

Abstract

The Turán number of a graph H, denoted by ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pk denote the path on k vertices and let mPk denote m disjoint copies of Pk. Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20 (2011) 837–853] determined the exact value of ex(n, kP) for large values of n. Yuan and Zhang [The Turán number of disjoint copies of paths, Discrete Math. 340 (2017) 132–139] completely determined the value of ex(n, kP 3) for all n, and also determined ex(n, Fm), where Fm is the disjoint union of m paths containing at most one odd path. They also determined the exact value of ex(n, P 3P 2 +1) for n ≥ 2 + 4. Recently, Bielak and Kieliszek [The Turán number of the graph 2P 5, Discuss. Math. Graph Theory 36 (2016) 683–694], Yuan and Zhang [Turán numbers for disjoint paths, arXiv:1611.00981v1] independently determined the exact value of ex(n, 2P 5). In this paper, we show that ex(n, 2P 7) = max{[n, 14, 7], 5n − 14} for all n ≥ 14, where [n, 14, 7] = (5n + 91 + r(r − 6))/2, n − 13 ≡ r (mod 6) and 0 ≤ r < 6.

Open access

H. Abdollahzadeh Ahangar, J. Amjadi, M. Chellali, S. Nazari-Moghaddam and S.M. Sheikholeslami

Abstract

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The minimum weight of a total Roman dominating function on a graph G is called the total Roman domination number of G. The total Roman reinforcement number rtR (G) of a graph G is the minimum number of edges that must be added to G in order to decrease the total Roman domination number. In this paper, we investigate the proper- ties of total Roman reinforcement number in graphs, and we present some sharp bounds for rtR (G). Moreover, we show that the decision problem for total Roman reinforcement is NP-hard for bipartite graphs.