The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_{G}(X)neq V(G)$. The association between the binding number and toughness is intricately interconnected, as both metrics function as pivotal indicators for quantifying the vulnerability of a graph. The Brouwer-Gu Theorem asserts that for any $d$-regular connected graph $G$, the toughness $t(G)$ always at least $frac{d}{lambda}-1$, where $lambda$ denotes the second largest absolute eigenvalue of the adjacency matrix. Inspired by the work of Brouwer and Gu, in this paper, we investigate $b(G)$ from spectral perspectives, and provide tight sufficient conditions in terms of the spectral radius of a graph $G$ to guarantee $b(G)geq r$. The study of the existence of $k$-factors in graphs is a classic problem in graph theory. Katerinis and Woodall state that every graph with order $ngeq 4k-6$ satisfying $b(G)geq 2$ contains a $k$-factor where $kgeq 2$. This leaves the following question: which $1$-binding graphs have a $k$-factor? In this paper, we also provide the spectral radius conditions of $1$-binding graphs to contain a perfect matching and a $2$-factor, respectively.
{"title":"Binding Number, $k$-Factor and Spectral Radius of Graphs","authors":"Dandan Fan, Huiqiu Lin","doi":"10.37236/12165","DOIUrl":"https://doi.org/10.37236/12165","url":null,"abstract":"The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_{G}(X)neq V(G)$. The association between the binding number and toughness is intricately interconnected, as both metrics function as pivotal indicators for quantifying the vulnerability of a graph. The Brouwer-Gu Theorem asserts that for any $d$-regular connected graph $G$, the toughness $t(G)$ always at least $frac{d}{lambda}-1$, where $lambda$ denotes the second largest absolute eigenvalue of the adjacency matrix. Inspired by the work of Brouwer and Gu, in this paper, we investigate $b(G)$ from spectral perspectives, and provide tight sufficient conditions in terms of the spectral radius of a graph $G$ to guarantee $b(G)geq r$. The study of the existence of $k$-factors in graphs is a classic problem in graph theory. Katerinis and Woodall state that every graph with order $ngeq 4k-6$ satisfying $b(G)geq 2$ contains a $k$-factor where $kgeq 2$. This leaves the following question: which $1$-binding graphs have a $k$-factor? In this paper, we also provide the spectral radius conditions of $1$-binding graphs to contain a perfect matching and a $2$-factor, respectively.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":" 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139789614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following: Let $a$, $b$ and $k$ be integers with $1leq aleq kleq bleq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $deg_{F}(x)in {a,b}$ for every red vertex $x$ and $deg_{F}(y)in {k,k+1}$ for every blue vertex $y$.
{"title":"Degree Factors with Red-Blue Coloring of Regular Graphs","authors":"Michitaka Furuya, Mikio Kano","doi":"10.37236/12299","DOIUrl":"https://doi.org/10.37236/12299","url":null,"abstract":"Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following: \u0000Let $a$, $b$ and $k$ be integers with $1leq aleq kleq bleq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $deg_{F}(x)in {a,b}$ for every red vertex $x$ and $deg_{F}(y)in {k,k+1}$ for every blue vertex $y$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":"23 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139870991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following: Let $a$, $b$ and $k$ be integers with $1leq aleq kleq bleq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $deg_{F}(x)in {a,b}$ for every red vertex $x$ and $deg_{F}(y)in {k,k+1}$ for every blue vertex $y$.
{"title":"Degree Factors with Red-Blue Coloring of Regular Graphs","authors":"Michitaka Furuya, Mikio Kano","doi":"10.37236/12299","DOIUrl":"https://doi.org/10.37236/12299","url":null,"abstract":"Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following: \u0000Let $a$, $b$ and $k$ be integers with $1leq aleq kleq bleq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $deg_{F}(x)in {a,b}$ for every red vertex $x$ and $deg_{F}(y)in {k,k+1}$ for every blue vertex $y$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":"24 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139811200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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