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The { K 1 , 1 , K 1 , 2 , ... , K 1 , k , (2 k + 1) } -factor and { K 1 , 2 , K 1 , 3 , K 5 } -factor of a graph are a spanning subgraph whose each component is an element of { K 1 , 1 , K 1 , 2 , ... , K 1 , k , (2 k + 1) } and { K 1 , 2 , K 1 , 3 , K 5 } , respectively, where (2 k + 1) is a special family of trees. In this paper, we obtain a sufficient condition in terms of tight toughness, isolated toughness and binding number bounds to guarantee the existence of a { K 1 , 1 , K 1 , 2 , ... , K 1 , k , (2 k + 1) } -factor and { K 1 , 2 , K 1 , 3 , K 5 } -factor for any graph. [ABSTRACT FROM AUTHOR]

Context: Weaning is a critical period for piglets, often associated with reduced growth performance and increased susceptibility to diseases. Dietary supplements such as fructo-oligosaccharides (FOS) and Bacillus licheniformis (BL) may improve piglet health and performance during this transition. Aims: To investigate the effects of dietary FOS and BL supplementation, individually and in combination, on growth performance, health status, nutrient digestibility, hematological parameters, and organ development in weaned piglets. Methods: In total, 240 weaned piglets were allocated into the following six treatments: control, 0.2% FOS, 0.4% FOS, BL, 0.2% FOS + BL, and 0.4% FOS + BL. Growth performance, morbidity rates, nutrient digestibility, blood parameters, and organ indices were evaluated over the experimental period. Key results: The 0.4% FOS + BL group showed significant improvements in feed conversion ratio, reduced morbidity rates, enhanced nutrient digestibility (crude protein, crude fiber, and phosphorus), increased total protein, albumin, and albumin:globulin ratio, reduced blood urea nitrogen, improved white blood cell and lymphocyte counts, and enhanced spleen and pancreatic organ indices. BL supplementation distinctively influenced liver-enzyme profiles, suggesting metabolic adaptation rather than hepatic stress. Blood urea nitrogen was reduced in BL and 0.4% FOS + BL groups. Conclusions: Dietary supplementation with 0.4% FOS and BL synergistically benefited weaned piglets by improving growth performance, enhancing feed efficiency, and reducing morbidity rates. This treatment also enhanced nutrient utilization, positively affected blood parameters, and promotes the development of spleen and pancreatic organs. Implications: The combination of 0.4% FOS and BL shows potential as an effective feed-additive strategy to enhance growth and immune responses in weaned piglets, potentially reducing the need for antibiotics and promoting sustainable pig production. Weaning stress can severely affect piglet growth, nutrient utilization and immune function. This study found that combining dietary supplements fructo-oligosaccharides (FOS) and Bacillus licheniformis (BL) significantly improved feed efficiency, nutrient digestibility, protein metabolism, immune-cell counts, and organ developments in weaned piglets. These findings have highlighted a sustainable, antibiotic-free strategy to support piglet development during the critical weaning transition, promoting healthier and more efficient pig-farming practices. [ABSTRACT FROM AUTHOR]

For a family of connected graphs ℱ , a spanning subgraph L of G is called an ℱ -factor if each component of L is isomorphic to a member of ℱ. In this paper, sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the { P 2 , C 3 , P 5 , (3) } -factor, P ≥ 2 -factor or { P 2 , C 2 i + 1 | i ≥ k } -factor for k ≥ 3 are obtained. [ABSTRACT FROM AUTHOR]