In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In this note, we simplify their proof, using an induction to replace the original analysis. This reduces the proof to one page and from the argument it is also easy to read that one can set the constant $K$ in the conjecture to $approx 3.998$, which could be the best value under the current method. Our argument also applies to the $epsilon$-version of the Park-Pham result, studied by Bell.
{"title":"A Short Proof of Kahn-Kalai Conjecture","authors":"Phuc Tran, Van Vu","doi":"10.37236/12266","DOIUrl":"https://doi.org/10.37236/12266","url":null,"abstract":"In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In this note, we simplify their proof, using an induction to replace the original analysis. This reduces the proof to one page and from the argument it is also easy to read that one can set the constant $K$ in the conjecture to $approx 3.998$, which could be the best value under the current method. Our argument also applies to the $epsilon$-version of the Park-Pham result, studied by Bell.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653103","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}
Let $mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P in mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.
{"title":"Hamiltonian Intervals in the Lattice of Binary Paths","authors":"I. Tasoulas, K. Manes, A. Sapounakis","doi":"10.37236/12144","DOIUrl":"https://doi.org/10.37236/12144","url":null,"abstract":"Let $mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P in mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139847886","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}
Let $mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P in mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.
{"title":"Hamiltonian Intervals in the Lattice of Binary Paths","authors":"I. Tasoulas, K. Manes, A. Sapounakis","doi":"10.37236/12144","DOIUrl":"https://doi.org/10.37236/12144","url":null,"abstract":"Let $mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P in mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139787966","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}
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":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139849542","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}
Let $G$ be a simple connected graph and $mu_1(G) geq mu_2(G) geq cdots geq mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $mu_{n-1}(G)+mu_{n-1}(overline{G})geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $mu_k(G)+mu_k(overline{G})geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $overline{G}$ is isomorphic to $K_{n-k}vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $kleq frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $ngeq 2c+8$.
{"title":"Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues","authors":"Wen-Jun Li, Ji-Ming Guo","doi":"10.37236/12008","DOIUrl":"https://doi.org/10.37236/12008","url":null,"abstract":"Let $G$ be a simple connected graph and $mu_1(G) geq mu_2(G) geq cdots geq mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $mu_{n-1}(G)+mu_{n-1}(overline{G})geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $mu_k(G)+mu_k(overline{G})geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $overline{G}$ is isomorphic to $K_{n-k}vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $kleq frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $ngeq 2c+8$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139788285","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}
Let $G$ be a simple connected graph and $mu_1(G) geq mu_2(G) geq cdots geq mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $mu_{n-1}(G)+mu_{n-1}(overline{G})geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $mu_k(G)+mu_k(overline{G})geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $overline{G}$ is isomorphic to $K_{n-k}vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $kleq frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $ngeq 2c+8$.
{"title":"Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues","authors":"Wen-Jun Li, Ji-Ming Guo","doi":"10.37236/12008","DOIUrl":"https://doi.org/10.37236/12008","url":null,"abstract":"Let $G$ be a simple connected graph and $mu_1(G) geq mu_2(G) geq cdots geq mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $mu_{n-1}(G)+mu_{n-1}(overline{G})geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $mu_{n-2}(G)+mu_{n-2}(overline{G})geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $mu_k(G)+mu_k(overline{G})geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $overline{G}$ is isomorphic to $K_{n-k}vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $kleq frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $ngeq 2c+8$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139848001","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}
As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.
{"title":"A Strengthening and an Efficient Implementation of Alon-Tarsi List Coloring Method","authors":"Zdenek Dvorák","doi":"10.37236/12058","DOIUrl":"https://doi.org/10.37236/12058","url":null,"abstract":"As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139848973","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}
The class of $k$-nearly finitary matroids for some natural number $k$ is a subclass of the class of nearly finitary matroids. A natural question is whether this inclusion is proper. We answer this question affirmatively by constructing a nearly finitary matroid that is not $k$-nearly finitary for any $k in mathbb{N}$.
{"title":"A Nearly Finitary Matroid that is not $k$-Nearly Finitary","authors":"Patrick Tam","doi":"10.37236/10467","DOIUrl":"https://doi.org/10.37236/10467","url":null,"abstract":"The class of $k$-nearly finitary matroids for some natural number $k$ is a subclass of the class of nearly finitary matroids. A natural question is whether this inclusion is proper. We answer this question affirmatively by constructing a nearly finitary matroid that is not $k$-nearly finitary for any $k in mathbb{N}$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139789097","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}
The class of $k$-nearly finitary matroids for some natural number $k$ is a subclass of the class of nearly finitary matroids. A natural question is whether this inclusion is proper. We answer this question affirmatively by constructing a nearly finitary matroid that is not $k$-nearly finitary for any $k in mathbb{N}$.
{"title":"A Nearly Finitary Matroid that is not $k$-Nearly Finitary","authors":"Patrick Tam","doi":"10.37236/10467","DOIUrl":"https://doi.org/10.37236/10467","url":null,"abstract":"The class of $k$-nearly finitary matroids for some natural number $k$ is a subclass of the class of nearly finitary matroids. A natural question is whether this inclusion is proper. We answer this question affirmatively by constructing a nearly finitary matroid that is not $k$-nearly finitary for any $k in mathbb{N}$.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139849062","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}
As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.
{"title":"A Strengthening and an Efficient Implementation of Alon-Tarsi List Coloring Method","authors":"Zdenek Dvorák","doi":"10.37236/12058","DOIUrl":"https://doi.org/10.37236/12058","url":null,"abstract":"As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.","PeriodicalId":509530,"journal":{"name":"The Electronic Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139789267","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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