On path-factor critical uniform graphs

Hongxia Liu ()
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Hongxia Liu: Yantai University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1222-1230

Abstract: Abstract A graph G is $$P_{\ge k}$$ P ≥ k -factor uniform if for arbitrary two distinct edges $$e_1$$ e 1 and $$e_2$$ e 2 of G, G contains a $$P_{\ge k}$$ P ≥ k -factor including $$e_1$$ e 1 and excluding $$e_2$$ e 2 . In this paper, we study the relationship between some graphic parameters and the existence of path factor with specific properties. Our main contributions are three-fold: firstly, we define the new concept of $$(P_{\ge k}, n)$$ ( P ≥ k , n ) -critical uniform graph, namely, a graph G is $$(P_{\ge k}, n)$$ ( P ≥ k , n ) -critical uniform if the graph $$G-V'$$ G - V ′ is $$P_{\ge k}$$ P ≥ k -factor uniform for any $$V'\subseteq V(G)$$ V ′ ⊆ V ( G ) with $$|V'|=n$$ | V ′ | = n . Secondly, the relationship between binding number and $$(P_{\ge k}, n)$$ ( P ≥ k , n ) -critical uniform graphs $$(k=2,3)$$ ( k = 2 , 3 ) is studied. Thirdly, we demonstrate the sharpness of the main results in this paper by constructing several graph classes. Furthermore, the relationship between other graph parameters and $$(P_{\ge k}, n)$$ ( P ≥ k , n ) -critical uniform graphs can be studied further.

Keywords: Binding number; $$P_{\ge 2}$$ P ≥ 2 -factor; $$P_{\ge 3}$$ P ≥ 3 -factor; $$(P_{\ge 2}; n)$$ ( P ≥ 2; n ) -critical uniform graph; $$(P_{\ge 3}; n)$$ ( P ≥ 3; n ) -critical uniform graph; 05C70; 05C38 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00428-9

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