Journal Details
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
Open Access

# Some Results on Path-Factor Critical Avoidable Graphs

###### Accepted: 03 Sep 2020
Journal Details
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English

A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. We write Pk = {Pi : ik}. Then a Pk-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a Pk-factor avoidable graph if for any eE(G), G admits a Pk-factor excluding e. A graph G is called a (Pk, n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, GQ is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P2, n)-factor critical avoidable graph if tough(G)>n+24tough\left( G \right) > {{n + 2} \over 4}; (ii) G is a (P3, n)-factor critical avoidable graph if tough(G)>n+12tough\left( G \right) > {{n + 1} \over 2}; (iii) G is a (P2, n)-factor critical avoidable graph if I(G)>n+23I\left( G \right) > {{n + 2} \over 3}; (iv) G is a (P3, n)-factor critical avoidable graph if I(G)>n+32I\left( G \right) > {{n + 3} \over 2}. Furthermore, we claim that these conditions are sharp.

#### MSC 2010

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