For a graph G and even integers b ⩾ a ⩾ 2, a spanning subgraph F of G such that a ⩽ degF (x) ⩽ b and degF (x) is even for all x ∈ V (F) is called an even [a, b]-factor of G. In this paper, we show that a 2-edge-connected graph G of order n has an even [2, b]-factor if [...] max degG (x),degG (y)⩾max 2n2+b,3 for any nonadjacent vertices x and y of G. Moreover, we show that for b ⩾ 3a and a > 2, there exists an infinite family of 2-edge-connected graphs G of order n with δ(G) ⩾ a such that G satisfies the condition [...] degG (x)+degG (y)>2ana+b for any nonadjacent vertices x and y of G, but has no even [a, b]-factors. In particular, the infinite family of graphs gives a counterexample to the conjecture of Matsuda on the existence of an even [a, b]-factor.
@article{bwmeta1.element.doi-10_7151_dmgt_1964,
author = {Shoichi Tsuchiya and Takamasa Yashima},
title = {A Degree Condition Implying Ore-Type Condition for Even [2,b]-Factors in Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {797-809},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1964}
}
Shoichi Tsuchiya; Takamasa Yashima. A Degree Condition Implying Ore-Type Condition for Even [2,b]-Factors in Graphs. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 797-809. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1964/