Graphs and Combinatorics最新文献

On Flipping Edge Sets in Unique Sink Orientations. 关于唯一Sink方向的翻转边集。

IF 0.6 4区数学

Graphs and Combinatorics Pub Date : 2025-01-01 Epub Date: 2025-04-25 DOI: 10.1007/s00373-025-02929-2

Michaela Borzechowski, Simon Weber

{"title":"On Flipping Edge Sets in Unique Sink Orientations.","authors":"Michaela Borzechowski, Simon Weber","doi":"10.1007/s00373-025-02929-2","DOIUrl":"https://doi.org/10.1007/s00373-025-02929-2","url":null,"abstract":"A unique sink orientation (USO) is an orientation of the n-dimensional hypercube graph such that every non-empty face contains a unique sink. We consider the only known connected flip graph on USOs. This flip graph is based on the following theorem due to Schurr: given any n-dimensional USO and any one dimension <math><mrow><mi>i</mi> <mo>∈</mo> <mo>[</mo> <mi>n</mi> <mo>]</mo></mrow> </math> , the set <math><msub><mi>E</mi> <mi>i</mi></msub> </math> of edges connecting vertices along dimension i can be decomposed into equivalence classes (so-called phases), such that flipping the direction of any <math><mrow><mi>S</mi> <mo>⊆</mo> <msub><mi>E</mi> <mi>i</mi></msub> </mrow> </math> yields another USO if and only if S is the union of some of these phases. In this paper we provide an algorithm to compute the phases of a given USO in <math><mrow><mi>O</mi> <mo>(</mo> <mi>n</mi> <mo>·</mo> <msup><mn>3</mn> <mi>n</mi></msup> <mo>)</mo></mrow> </math> time, significantly improving upon the previously known <math><mrow><mi>O</mi> <mo>(</mo> <mi>n</mi> <mo>·</mo> <msup><mn>4</mn> <mi>n</mi></msup> <mo>)</mo></mrow> </math> trivial algorithm. We also show that the phase containing a given edge can be flipped using only poly(n) space additional to the space required to store the USO. We contrast this by showing that given a boolean circuit of size poly(n) succinctly encoding an n-dimensional USO, it is <math><mi>PSPACE</mi></math> -complete to determine whether two given edges are in the same phase. Finally, we also prove some new results on the structure of phases.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"41 3","pages":"64"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031957/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144007340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An Efficient Algorithm to Compute the Toughness in Graphs with Bounded Treewidth 计算有界树宽图中韧性的高效算法

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-22 DOI: 10.1007/s00373-024-02828-y

Gyula Y. Katona, Humara Khan

引用次数: 0

On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree 论小度无爪图中局部公平着色的复杂性

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-16 DOI: 10.1007/s00373-024-02826-0

Zuosong Liang

{"title":"On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree","authors":"Zuosong Liang","doi":"10.1007/s00373-024-02826-0","DOIUrl":"https://doi.org/10.1007/s00373-024-02826-0","url":null,"abstract":"An equitable k-partition ((kge 2)) of a vertex set S is a partition of S into k subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A local-equitable k-coloring ((kge 2)) of G is an assignment of k colors to the vertices of G such that, for every maximal clique H of G, the coloring on H forms an equitable k-partition of H. Local-equitable coloring of graphs is a generalization of the proper vertex coloring of graphs and also a stronger version of clique-coloring of graphs. Claw-free graphs with maximum degree four are proved to be 2-clique-colorable [Discrete Math. Theoret. Comput. Sci. 11 (2) (2009), 15–24] but not necessary local-equitably 2-colorable. In this paper, given a claw-free graph G with maximum degree at most four, we present a linear time algorithm to give a local-equitable 2-coloring of G or decide that G is not local-equitably 2-colorable. As a corollary, we get that claw-free perfect graphs with maximum degree at most four are local-equitably 2-colorable.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

New Tools to Study 1-11-Representation of Graphs 研究 1-11 的新工具--图形表示法

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-14 DOI: 10.1007/s00373-024-02825-1

Mikhail Futorny, Sergey Kitaev, Artem Pyatkin

{"title":"New Tools to Study 1-11-Representation of Graphs","authors":"Mikhail Futorny, Sergey Kitaev, Artem Pyatkin","doi":"10.1007/s00373-024-02825-1","DOIUrl":"https://doi.org/10.1007/s00373-024-02825-1","url":null,"abstract":"The notion of a k-11-representable graph was introduced by Jeff Remmel in 2017 and studied by Cheon et al. in 2019 as a natural extension of the extensively studied notion of word-representable graphs, which are precisely 0-11-representable graphs. A graph G is k-11-representable if it can be represented by a word w such that for any edge (resp., non-edge) xy in G the subsequence of w formed by x and y contains at most k (resp., at least (k+1)) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that any graph is 2-11-representable, while it is unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs. In this paper, we introduce new tools for studying 1-11-representation of graphs. We apply them for establishing 1-11-representation of Chvátal graph, Mycielski graph, split graphs, and graphs whose vertices can be partitioned into a comparability graph and an independent set.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Vertex Arboricity of 1-Planar Graphs 单平面图形的顶点有性

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-08 DOI: 10.1007/s00373-024-02820-6

Dongdong Zhang, Juan Liu, Yongjie Li, Hehua Yang

引用次数: 0

Maximal Colourings for Graphs 图形的最大着色

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-07 DOI: 10.1007/s00373-024-02823-3

Raffaella Mulas

引用次数: 0

Independence Number and Maximal Chromatic Polynomials of Connected Graphs 连通图的独立数和最大色度多项式

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-07 DOI: 10.1007/s00373-024-02824-2

Shude Long, Junliang Cai

引用次数: 0

Normalized Laplacian Eigenvalues of Hypergraphs 超图的归一化拉普拉奇特征值

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-08-03 DOI: 10.1007/s00373-024-02815-3

Leyou Xu, Bo Zhou

引用次数: 0

On the Sequence with Fewer Subsequence Sums in Finite Abelian Groups 论有限阿贝尔群中具有较少后继和的序列

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-07-31 DOI: 10.1007/s00373-024-02818-0

Jiangtao Peng, Yue Sun

引用次数: 0

Gallai-Ramsey Multiplicity for Rainbow Small Trees 彩虹小树的加莱-拉姆齐多重性

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-07-31 DOI: 10.1007/s00373-024-02819-z

Xueliang Li, Yuan Si

引用次数: 0