{"title":"具有所有 (a, b) 奇偶因子的图的度条件","authors":"Hao-dong Liu, Hong-liang Lu","doi":"10.1007/s10255-024-1090-y","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>a</i> and <i>b</i> be positive integers such that <i>a</i> ≤ <i>b</i> and <i>a</i> ≡ <i>b</i> (mod 2). We say that <i>G</i> has all (<i>a, b</i>)-parity factors if <i>G</i> has an <i>h</i>-factor for every function <i>h</i>: <i>V</i>(<i>G</i>) → {<i>a, a</i> + 2, ⋯, <i>b</i> − 2, <i>b</i>} with <i>b</i>∣<i>V</i>(<i>G</i>)∣ even and <i>h</i>(<i>v</i>) ≡ <i>b</i> (mod 2) for all <i>v</i> ∈ <i>V</i>(<i>G</i>). In this paper, we prove that every graph <i>G</i> with <i>n</i> ≥ 2(<i>b</i> + 1)(<i>a</i> + <i>b</i>) vertices has all (<i>a, b</i>)-parity factors if <i>δ</i>(<i>G</i>) ≥ (<i>b</i><sup>2</sup> − <i>b</i>)/<i>a</i>, and for any two nonadjacent vertices <span>\\(u,\\,v\\, \\in \\,V\\,(G),\\,\\max \\{{d_G}(u),\\,{d_G}(v)\\} \\, \\ge {{bn} \\over {a + b}}\\)</span>. Moreover, we show that this result is best possible in some sense.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Degree Condition for Graphs Having All (a, b)-parity Factors\",\"authors\":\"Hao-dong Liu, Hong-liang Lu\",\"doi\":\"10.1007/s10255-024-1090-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>a</i> and <i>b</i> be positive integers such that <i>a</i> ≤ <i>b</i> and <i>a</i> ≡ <i>b</i> (mod 2). We say that <i>G</i> has all (<i>a, b</i>)-parity factors if <i>G</i> has an <i>h</i>-factor for every function <i>h</i>: <i>V</i>(<i>G</i>) → {<i>a, a</i> + 2, ⋯, <i>b</i> − 2, <i>b</i>} with <i>b</i>∣<i>V</i>(<i>G</i>)∣ even and <i>h</i>(<i>v</i>) ≡ <i>b</i> (mod 2) for all <i>v</i> ∈ <i>V</i>(<i>G</i>). In this paper, we prove that every graph <i>G</i> with <i>n</i> ≥ 2(<i>b</i> + 1)(<i>a</i> + <i>b</i>) vertices has all (<i>a, b</i>)-parity factors if <i>δ</i>(<i>G</i>) ≥ (<i>b</i><sup>2</sup> − <i>b</i>)/<i>a</i>, and for any two nonadjacent vertices <span>\\\\(u,\\\\,v\\\\, \\\\in \\\\,V\\\\,(G),\\\\,\\\\max \\\\{{d_G}(u),\\\\,{d_G}(v)\\\\} \\\\, \\\\ge {{bn} \\\\over {a + b}}\\\\)</span>. Moreover, we show that this result is best possible in some sense.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1090-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1090-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
设 a 和 b 为正整数，且 a≤b 和 a≡b (mod 2)。如果对于每个函数 h，G 都有一个 h 因子，那么我们就说 G 具有所有 (a, b) 奇偶因子：V(G)→{a，a + 2，⋯，b - 2，b}，其中 b∣V(G)∣ 偶数，且对于所有 v∈V(G) ，h(v) ≡ b（mod 2）。在本文中，我们将证明，如果 δ(G) ≥ (b2 - b)/a, 并且对于任意两个非相邻顶点 \(u,\,v\, \in \,V\,(G),\,\max \{{d_G}(u),\,{d_G}(v)\} ，则具有 n≥ 2(b + 1)(a + b) 个顶点的每个图 G 都具有所有（a, b）奇偶因子。\ge {{bn}\over {a + b}}）。此外，我们还证明了这一结果在某种意义上是最好的。
A Degree Condition for Graphs Having All (a, b)-parity Factors
Let a and b be positive integers such that a ≤ b and a ≡ b (mod 2). We say that G has all (a, b)-parity factors if G has an h-factor for every function h: V(G) → {a, a + 2, ⋯, b − 2, b} with b∣V(G)∣ even and h(v) ≡ b (mod 2) for all v ∈ V(G). In this paper, we prove that every graph G with n ≥ 2(b + 1)(a + b) vertices has all (a, b)-parity factors if δ(G) ≥ (b2 − b)/a, and for any two nonadjacent vertices \(u,\,v\, \in \,V\,(G),\,\max \{{d_G}(u),\,{d_G}(v)\} \, \ge {{bn} \over {a + b}}\). Moreover, we show that this result is best possible in some sense.
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