# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## GALLAI'S INNEQUALITY FOR CRITICAL GRAPHS OF REDUCIBLE HEREDITARY PROPERTIES

 Peter Mihók Mathematical Institute of Slovak Academy of Sciences Gresákova 6, 040 01 Košice, Slovakia and Faculty of Economics, Technical University B. Nemcovej 32, 040 01 Košice, Slovakia e-mail: mihokp@tuke.sk Riste Skrekovski Departement of Mathematics University of Ljubljana Jadranska 19, 1111 Ljubljana, Slovenia e-mail: skreko@fmf.uni-lj.si

## Abstract

In this paper Gallai's inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let P1, P2, …,Pk (k ≥ 2) be additive induced-hereditary properties, ℜ = P1 º P2 º… ºPk and δ = ∑i = 1k δ(Pi). Suppose that G is an ℜ -critical graph with n vertices and m edges. Then 2m ≥ δn + [(δ−2)/(δ2+2δ−2)] n + [(2δ)/(δ2+2δ−2)] unless ℜ = O2 or G = Kδ+1. The generalization of Gallai's inequality for P-choice critical graphs is also presented.

Keywords: additive induced-hereditary property of graphs, reducible property of graphs, critical graph, Gallai's Theorem.

2000 Mathematics Subject Classification: 05C15, 05C75.

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