{"title":"χ-boundedness and related problems on graphs without long induced paths: A survey","authors":"Arnab Char, T. Karthick","doi":"10.1016/j.dam.2024.12.014","DOIUrl":"10.1016/j.dam.2024.12.014","url":null,"abstract":"<div><div>Given a hereditary class of graphs <span><math><mi>G</mi></math></span>, a function <span><math><mrow><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></mrow></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mi>x</mi></mrow></math></span>, for all <span><math><mrow><mi>x</mi><mo>∈</mo><mi>N</mi></mrow></math></span> is a <span><math><mi>χ</mi></math></span><em>-binding function</em> for <span><math><mi>G</mi></math></span> if <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, for each <span><math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math></span>. The class <span><math><mi>G</mi></math></span> is called <span><math><mi>χ</mi></math></span><em>-bounded</em> if there exists a <span><math><mi>χ</mi></math></span>-binding function for <span><math><mi>G</mi></math></span>. The class of graphs without long induced paths is well-studied and also received a wide recognition among the researchers for the past few decades. Here we present a survey on <span><math><mi>χ</mi></math></span>-boundedness for some classes of graphs without long induced paths by giving more attention to structure/decomposition theorems which led to such results, and we discuss some related well-known conjectures and other problems of interest including algorithmic complexity, and their connections to <span><math><mi>χ</mi></math></span>-boundedness.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 99-119"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On saturated non-covered graphs","authors":"Jinqiu Zhou, Yuefen Cao, Weigen Yan","doi":"10.1016/j.dam.2024.12.013","DOIUrl":"10.1016/j.dam.2024.12.013","url":null,"abstract":"<div><div>A connected simple graph <span><math><mi>G</mi></math></span> of order <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span> with edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is a <em>matching covered graph</em> if it has at least one edge and each edge of <span><math><mi>G</mi></math></span> is contained in some perfect matching of <span><math><mi>G</mi></math></span>. Define a graph <span><math><mi>G</mi></math></span> to be <em>saturated non-covered</em> if it is not matching covered, but <span><math><mrow><mi>G</mi><mo>+</mo><mi>e</mi></mrow></math></span> for any <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the complement of <span><math><mi>G</mi></math></span>, is a matching covered graph. In this paper, we obtain a characterization of saturated non-covered graphs. Furthermore, we prove that <span><math><mi>G</mi></math></span> of order <span><math><mrow><mn>2</mn><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span> is matching covered if one of the following statements holds: (1) the number of spanning trees of <span><math><mi>G</mi></math></span> is either more than 324 for <span><math><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow></math></span> or more than <span><math><mrow><mn>4</mn><mi>n</mi><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msup></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≠</mo><mn>3</mn></mrow></math></span>; (2) the Wiener index of <span><math><mi>G</mi></math></span> is less than <span><math><mrow><mn>2</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>; (3) the Kirchhoff index of <span><math><mi>G</mi></math></span> is less than <span><math><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 53-59"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jiangdong Ai , Qiming Dai , Qiwen Guo , Yingqi Hu , Changxin Wang
{"title":"A new perspective from hypertournaments to tournaments","authors":"Jiangdong Ai , Qiming Dai , Qiwen Guo , Yingqi Hu , Changxin Wang","doi":"10.1016/j.dam.2024.12.010","DOIUrl":"10.1016/j.dam.2024.12.010","url":null,"abstract":"<div><div>A <span><math><mi>k</mi></math></span>-hypertournament <span><math><mi>H</mi></math></span> on <span><math><mi>n</mi></math></span> vertices is a pair <span><math><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></math></span> for <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></math></span>, where <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is a set of vertices, and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is a set of all possible <span><math><mi>k</mi></math></span>-tuples of vertices, such that for any <span><math><mi>k</mi></math></span>-subset <span><math><mi>S</mi></math></span> of <span><math><mi>V</mi></math></span>, <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> contains exactly one of the <span><math><mrow><mi>k</mi><mo>!</mo></mrow></math></span> possible permutations of <span><math><mi>S</mi></math></span>. In this paper, we investigate the relationship between a hyperdigraph and its corresponding normal digraph. Particularly, drawing on a result from Gutin and Yeo (1997), we establish an intrinsic relationship between a strong <span><math><mi>k</mi></math></span>-hypertournament and a strong tournament, which enables us to provide an alternative (more straightforward and concise) proof for some previously known results and get some new results.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 136-142"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Further acceleration in computing the gap greedy spanner: An empirical approach","authors":"Hosein Salami, Mostafa Nouri-Baygi","doi":"10.1016/j.dam.2024.12.003","DOIUrl":"10.1016/j.dam.2024.12.003","url":null,"abstract":"<div><div>Consider a set <span><math><mi>V</mi></math></span> of <span><math><mi>n</mi></math></span> points on the plane and a weighted geometric network <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> where the weight of each <span><math><mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><mo>∈</mo><mi>E</mi></mrow></math></span> is equal to the Euclidean distance between the endpoints of the edge (<span><math><mrow><mo>|</mo><mi>p</mi><mi>q</mi><mo>|</mo></mrow></math></span>). Given a constant <span><math><mrow><mi>t</mi><mo>></mo><mn>1</mn></mrow></math></span>, a spanning subgraph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> is said to be a <span><math><mi>t</mi></math></span>-spanner, or simply a spanner, if for any pair of nodes <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span> in <span><math><mi>G</mi></math></span> there exists a path between <span><math><mi>p</mi></math></span> and <span><math><mi>q</mi></math></span> whose length is at most <span><math><mi>t</mi></math></span> times their distance in <span><math><mi>G</mi></math></span>.</div><div><em>Gap greedy</em> spanner, proposed by Arya and Smid, is a light weight and bounded degree spanner in which a pair of points <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span> is guaranteed to have a <span><math><mi>t</mi></math></span>-path, if there exists at least one edge with some special criteria in the spanner. In our previous work, we introduced an algorithm with linear space complexity that takes <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> time to construct this spanner, utilizing well-separated pair decomposition. However, empirical results from experiments revealed that the performance of the algorithm is significantly dependent on the number of well-separated pairs obtained. In this paper, to mitigate the construction dependency on the number of well-separated pairs, we confine its usage to the processing of a small subset of point pairs. This limitation stems from the observation that a substantial portion of the edges in this spanner have small sizes. Consequently, we first empirically highlight the prevalence of the small edge size characteristic within the spanner. Subsequently, we propose an algorithm for computing this spanner that, based on the experimental results, exhibits a higher computational speed in generating the gap greedy spanner in most cases.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 33-52"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Distinguishing graphs via cycles","authors":"Nina Klobas , Matjaž Krnc","doi":"10.1016/j.dam.2024.11.034","DOIUrl":"10.1016/j.dam.2024.11.034","url":null,"abstract":"<div><div>In this paper, we employ the cycle regularity parameter to devise efficient recognition algorithms for three highly symmetric graph families: folded cubes, <span><math><mi>I</mi></math></span>-graphs, and double generalized Petersen graphs.</div><div>For integers <span><math><mrow><mi>ℓ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>m</mi></mrow></math></span> a simple graph is <span><math><mrow><mo>[</mo><mi>ℓ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>m</mi><mo>]</mo></mrow></math></span>-cycle regular if every path of length <span><math><mi>ℓ</mi></math></span> belongs to exactly <span><math><mi>λ</mi></math></span> different cycles of length <span><math><mi>m</mi></math></span>. We identify all <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>λ</mi><mo>,</mo><mn>8</mn><mo>]</mo></mrow></math></span>-cycle regular <span><math><mi>I</mi></math></span>-graphs and all <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>λ</mi><mo>,</mo><mn>8</mn><mo>]</mo></mrow></math></span>-cycle regular double generalized Petersen graphs. For <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span> we show that a folded cube <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> is <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>]</mo></mrow></math></span>, <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mn>4</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>12</mn><mi>n</mi><mo>+</mo><mn>8</mn><mo>,</mo><mn>6</mn><mo>]</mo></mrow></math></span> and <span><math><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>8</mn><mo>,</mo><mn>6</mn><mo>]</mo></mrow></math></span>-cycle regular, and identify the corresponding exceptional values of cycle regularity for <span><math><mrow><mi>n</mi><mo><</mo><mn>7</mn></mrow></math></span>. As a consequence, we describe a linear recognition algorithm for double generalized Petersen graphs, an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>E</mi><mo>|</mo></mrow><mo>log</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span> recognition algorithm for the family of folded cubes, and an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> recognition algorithm for <span><math><mi>I</mi></math></span>-graphs.</div><div>We believe the structural observations and methods used in the paper are of independent interest and could be used to solve other algorithmic problems. The results of this paper have been presented at COCOON 2021.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 74-98"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A lower bound on the constant in the Fourier min-entropy/influence conjecture","authors":"Aniruddha Biswas, Palash Sarkar","doi":"10.1016/j.dam.2024.12.011","DOIUrl":"10.1016/j.dam.2024.12.011","url":null,"abstract":"<div><div>We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be <span><math><mrow><mn>128</mn><mo>/</mo><mn>45</mn><mo>≈</mo><mn>2</mn><mo>.</mo><mn>8444</mn></mrow></math></span> which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, <span><math><mrow><mn>128</mn><mo>/</mo><mn>45</mn></mrow></math></span> is also presently the best known lower bound on the universal constant of the Fourier min-entropy/influence conjecture.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 23-32"},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Facial matchings in plane graphs","authors":"Július Czap , Stanislav Jendrol’","doi":"10.1016/j.dam.2024.12.015","DOIUrl":"10.1016/j.dam.2024.12.015","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a plane graph. Two edges of <span><math><mi>G</mi></math></span> are facially adjacent if they are consecutive on the facial walk of a face of <span><math><mi>G</mi></math></span>. A set of edges <span><math><mi>M</mi></math></span> in <span><math><mi>G</mi></math></span> is a (facial) matching if no two edges of <span><math><mi>M</mi></math></span> are (facially) adjacent in <span><math><mi>G</mi></math></span>. Matchings in graphs are well studied, since finding large matchings in graphs has many applications.</div><div>In this note we investigate facial matchings and present tight estimations on sizes of maximal facial matchings, maximum facial matchings, and perfect facial matchings in general plane graphs, bridgeless plane graphs, and plane triangulations.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 16-22"},"PeriodicalIF":1.0,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Maximum locally irregular induced subgraphs via minimum irregulators","authors":"Foivos Fioravantes , Nikolaos Melissinos , Theofilos Triommatis","doi":"10.1016/j.dam.2024.12.007","DOIUrl":"10.1016/j.dam.2024.12.007","url":null,"abstract":"<div><div>If a graph <span><math><mi>G</mi></math></span> is such that no two of its adjacent vertices have the same degree, we say that <span><math><mi>G</mi></math></span> is <em>locally irregular</em>. In this work we introduce and study the problem of finding a largest locally irregular induced subgraph of a given graph <span><math><mi>G</mi></math></span>. Equivalently, given a graph <span><math><mi>G</mi></math></span>, find a subset <span><math><mi>S</mi></math></span> of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with minimum order, such that deleting the vertices of <span><math><mi>S</mi></math></span> from <span><math><mi>G</mi></math></span> results in a locally irregular graph; we denote with <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> the order of such a set <span><math><mi>S</mi></math></span>. We first examine some easy graph families, namely paths, cycles, complete bipartite and complete graphs. However, we show that the decision version of the introduced problem is <span><math><mi>NP</mi></math></span>-Complete, even for restricted families of graphs, such as subcubic planar bipartite, or cubic bipartite graphs. We then show that we cannot even approximate an optimal solution within a ratio of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mi>n</mi></math></span> is the order of the graph, unless <span><math><mi>P</mi></math></span>=<span><math><mi>NP</mi></math></span>, even when the input graph is bipartite.</div><div>Then, looking for more positive results, we turn our attention towards computing <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> through the lens of parameterised complexity. In particular, we provide two algorithms that compute <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, each one considering different parameters. The first one considers the size of the solution <span><math><mi>k</mi></math></span> and the maximum degree <span><math><mi>Δ</mi></math></span> of <span><math><mi>G</mi></math></span> with running time <span><math><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>, while the second one considers the treewidth <span><math><mrow><mi>t</mi><mi>w</mi></mrow></math></span> and <span><math><mi>Δ</mi></math></span> of <span><math><mi>G</mi></math></span>, and has running time <span><math><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mn>3</mn><mi>t</mi><mi>w</mi></mrow></msup><msup><mrow><mi","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"363 ","pages":"Pages 168-189"},"PeriodicalIF":1.0,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Monochromatic graph decompositions inspired by anti-Ramsey colorings","authors":"Yair Caro , Zsolt Tuza","doi":"10.1016/j.dam.2024.12.009","DOIUrl":"10.1016/j.dam.2024.12.009","url":null,"abstract":"<div><div>We consider coloring problems inspired by the theory of anti-Ramsey /rainbow colorings that we generalize to a far extent.</div><div>Let <span><math><mi>F</mi></math></span> be a hereditary family of graphs; i.e., if <span><math><mrow><mi>H</mi><mo>∈</mo><mi>F</mi></mrow></math></span> and <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>H</mi></mrow></math></span> then also <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>F</mi></mrow></math></span>. For a graph <span><math><mi>G</mi></math></span> and any integer <span><math><mrow><mi>n</mi><mo>≥</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span>, let <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>|</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> denote the smallest number <span><math><mi>k</mi></math></span> of colors such that any edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with at least <span><math><mi>k</mi></math></span> colors forces a copy of <span><math><mi>G</mi></math></span> in which each color class induces a member of <span><math><mi>F</mi></math></span>.</div><div>The case <span><math><mrow><mi>F</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></mrow></mrow></math></span> is the notorious anti-Ramsey rainbow coloring problem introduced by Erdős, Simonovits and Sós in 1973.</div><div>Using the <span><math><mi>F</mi></math></span>-deck of <span><math><mi>G</mi></math></span>, <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>|</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>H</mi><mo>:</mo><mi>H</mi><mo>=</mo><mi>G</mi><mo>−</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>D</mi><mo>∈</mo><mi>F</mi><mo>}</mo></mrow></mrow></math></span>, we define <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>F</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>min</mo><mrow><mo>{</mo><mi>χ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>:</mo><mi>H</mi><mo>∈</mo><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>|</mo><mi>F</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>.</div><div>The main theorem we prove is: Suppose <span><math><mi>F</mi></math></span> is a hereditary family of graphs, and let <span><math><mi>G</mi></math></span> be a graph not a member of <span><math><mi>F</mi></math></span>. (1) If <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>F</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></mrow></math></span>, then <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>|</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mspace></mspace><mi>ex","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"363 ","pages":"Pages 190-200"},"PeriodicalIF":1.0,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jean-Luc Baril , Sergey Kirgizov , José L. Ramírez , Diego Villamizar
{"title":"The combinatorics of Motzkin polyominoes","authors":"Jean-Luc Baril , Sergey Kirgizov , José L. Ramírez , Diego Villamizar","doi":"10.1016/j.dam.2024.12.002","DOIUrl":"10.1016/j.dam.2024.12.002","url":null,"abstract":"<div><div>A word <span><math><mrow><mi>w</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> over the set of positive integers is a Motzkin word whenever <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mstyle><mi>1</mi></mstyle></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> for <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></math></span>. It can be associated to a <span><math><mi>n</mi></math></span>-column Motzkin polyomino whose <span><math><mi>i</mi></math></span>-th column contains <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> cells, and all columns are bottom-justified. We reveal bijective connections between Motzkin paths, restricted Catalan words, primitive Łukasiewicz paths, and Motzkin polyominoes. Using the aforementioned bijections together with classical one-to-one correspondence with Dyck paths avoiding <span><math><mrow><mi>U</mi><mi>D</mi><mi>U</mi></mrow></math></span>s, we provide generating functions with respect to the length, area, semiperimeter, value of the last symbol, and number of interior points of Motzkin polyominoes. We give asymptotics and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points over all Motzkin polyominoes of a given length. We also present and prove an engaging trinomial relation concerning the number of cells lying at different levels and first terms of the expanded <span><math><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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