Discrete Applied Mathematics最新文献

{"title":"On platypus graphs and the Steiner–Deogun property","authors":"Carol T. Zamfirescu","doi":"10.1016/j.dam.2025.06.048","DOIUrl":"10.1016/j.dam.2025.06.048","url":null,"abstract":"<div><div>A <em>platypus</em> is a non-hamiltonian graph in which every vertex-deleted subgraph is traceable. We prove a series of results on platypus graphs. For instance, although there are planar platypuses and bipartite platypuses, it is not known whether there is a planar bipartite platypus. Motivated by this question, we show that every tree is an induced subgraph of some planar platypus. On the other hand, there exists an infinite family of planar graphs each member of which is not an induced subgraph of any planar platypus. Throughout the article we point out connections between platypus graphs and graphs having the Steiner–Deogun property, as defined by Kratsch, Lehel, and Müller.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 87-94"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557045","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":"Characterization of minimally t-tough, 2K2-free graphs for 1<t≤2","authors":"Hui Ma,&nbsp;Xiaomin Hu,&nbsp;Weihua Yang","doi":"10.1016/j.dam.2025.06.045","DOIUrl":"10.1016/j.dam.2025.06.045","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is minimally <span><math><mi>t</mi></math></span>-tough if the toughness of <span><math><mi>G</mi></math></span> is exactly <span><math><mi>t</mi></math></span> and the removal of any edge decreases the toughness. Kriesell’s conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell’s conjecture that every minimally <span><math><mi>t</mi></math></span>-tough graph has a vertex of degree <span><math><mrow><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></mrow></math></span> for any positive rational number <span><math><mi>t</mi></math></span>. We have confirmed Kriesell’s conjecture for <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graphs by showing that every minimally 1-tough, <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>. In this paper, we prove that for <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>t</mi><mo>≤</mo><mn>2</mn></mrow></math></span>, every minimally <span><math><mi>t</mi></math></span>-tough, <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph on at least 14 vertices has a vertex of degree <span><math><mrow><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></mrow></math></span> by characterizing the structure of these graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 43-50"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550025","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 near-optimal kernel for a coloring problem","authors":"Ishay Haviv,&nbsp;Dror Rabinovich","doi":"10.1016/j.dam.2025.06.065","DOIUrl":"10.1016/j.dam.2025.06.065","url":null,"abstract":"<div><div>For a fixed integer <span><math><mi>q</mi></math></span>, the <span><math><mrow><mi>q</mi><mtext>− Coloring</mtext></mrow></math></span> problem asks to decide if a given graph has a vertex coloring with <span><math><mi>q</mi></math></span> colors such that no two adjacent vertices receive the same color. In a series of papers, it has been shown that for every <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, the <span><math><mrow><mi>q</mi><mtext>− Coloring</mtext></mrow></math></span> problem parameterized by the vertex cover number <span><math><mi>k</mi></math></span> admits a kernel of bit-size <span><math><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, but admits no kernel of bit-size <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>ɛ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> unless <span><math><mrow><mi>NP</mi><mo>⊆</mo><mi>coNP/poly</mi></mrow></math></span> (Jansen and Kratsch, 2013; Jansen and Pieterse, 2019). In 2020, Schalken proposed the question of the kernelizability of the <span><math><mrow><mi>q</mi><mtext>− Coloring</mtext></mrow></math></span> problem parameterized by the number <span><math><mi>k</mi></math></span> of vertices whose removal results in a disjoint union of edges and isolated vertices. He proved that for every <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, the problem admits a kernel of bit-size <span><math><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, but admits no kernel of bit-size <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>−</mo><mn>3</mn><mo>−</mo><mi>ɛ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> unless <span><math><mrow><mi>NP</mi><mo>⊆</mo><mi>coNP/poly</mi></mrow></math></span>. He further proved that for <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></mrow></math></span> the problem admits a near-optimal kernel of bit-size <span><math><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and asked whether such a kernel is achievable for all integers <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. In this short paper, we settle this question in the affirmative.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 66-73"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550028","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":"Robustness of reliability bounds for (n,k)-star networks","authors":"Hao Li ,&nbsp;Hongwei Qiao ,&nbsp;Eminjan Sabir","doi":"10.1016/j.dam.2025.06.064","DOIUrl":"10.1016/j.dam.2025.06.064","url":null,"abstract":"<div><div>The sub-network reliability, SNR for short, of a network is the probability that a healthy sub-network of a certain scale is operational in the network when faults occurs. It is surprisingly found that some networks’ SNR are different even though they have the same cardinality of processors. Hereby, we conduct comparative analysis for a pair of distinct topologies of <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-star networks <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> with same cardinality of processors to assess the robustness of their SNR bounds. We demonstrate analytically that SNR in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> is proportional to <span><math><mi>n</mi></math></span> when sub-networks are of the same cardinality of processors. This study presents a theoretical framework for selecting the more reliable topology among <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>s of the same cardinality of processors. We also conduct numerical simulations to verify our theoretical findings.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 51-65"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550026","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":"The list r-hued coloring of Halin graph","authors":"Shudan Lu ,&nbsp;Fengxia Liu ,&nbsp;Hong-Jian Lai","doi":"10.1016/j.dam.2025.06.044","DOIUrl":"10.1016/j.dam.2025.06.044","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi></math></span> be a list assignment of colors available for vertices of a graph <span><math><mi>G</mi></math></span>. An <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring of a graph <span><math><mi>G</mi></math></span> is a proper coloring <span><math><mi>c</mi></math></span> such that every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> receives at least <span><math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>,</mo><mi>r</mi><mo>}</mo></mrow></mrow></math></span> colors in its neighbors and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. The list <span><math><mi>r</mi></math></span>-hued chromatic number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span>, is the smallest <span><math><mi>k</mi></math></span> such that for each list assignment <span><math><mi>L</mi></math></span> satisfying <span><math><mrow><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span>, for any <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mi>G</mi></math></span> has an <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring. An <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring is called an <span><math><mi>r</mi></math></span>-hued <span><math><mi>k</mi></math></span>-coloring if <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mi>G</mi></math></span> be a Halin graph. We determine the upper bounds of the list <span><math><mi>r</mi></math></span>-hued chromatic number of Halin graphs when <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>=</mo><mi>Δ</mi></mrow></math></span>. This improves former results on 2-hued coloring of Halin graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 32-42"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550009","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":"Spectral analysis of eccentricity matrix for one-point-corona of graphs","authors":"Smrati Pandey,&nbsp;Lavanya Selvaganesh","doi":"10.1016/j.dam.2025.06.042","DOIUrl":"10.1016/j.dam.2025.06.042","url":null,"abstract":"<div><div>Among various graph operations, the corona product stands out as one of the well-known and extensively analyzed due to its elegant structural properties. Over the time, numerous variants of the corona operation have been introduced and are widely studied. In this paper, we investigate the spectral properties of the eccentricity matrix for one such variant. Let <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> be two connected graphs on <span><math><mi>m</mi></math></span> and <span><math><mi>n</mi></math></span> vertices, respectively. Let <span><math><mi>H</mi></math></span> be rooted at a designated vertex <span><math><mi>z</mi></math></span>. One-point-corona, <span><math><mrow><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>z</mi></mrow></msub><mi>H</mi></mrow></math></span> is obtained by taking one copy of <span><math><mi>H</mi></math></span> for each vertex of <span><math><mi>G</mi></math></span> and joining the root vertex <span><math><mi>z</mi></math></span> in each copy of <span><math><mi>H</mi></math></span> to the corresponding vertex in <span><math><mi>G</mi></math></span> by an edge. In this article, we study the eccentricity matrix, <span><math><mi>ɛ</mi></math></span>, of one-point-corona of two graphs. First, we characterize the irreducibility of <span><math><mrow><mi>ɛ</mi><mrow><mo>(</mo><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>z</mi></mrow></msub><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. We prove this result by demonstrating the connectedness of the direct product of two graphs when one of them has a self-loop. Under the assumption that <span><math><mi>G</mi></math></span> is self-centered, we investigate the <span><math><mi>ɛ</mi></math></span>-spectrum for <span><math><mrow><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>z</mi></mrow></msub><mi>H</mi></mrow></math></span> and identify the collection of <span><math><mi>ɛ</mi></math></span>-cospectral graphs and extremal graphs. In this process of finding the extremal graphs in terms of their <span><math><mi>ɛ</mi></math></span>-spectral radius, we establish a stronger result of organizing the graphs, <span><math><mrow><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>z</mi></mrow></msub><mi>H</mi></mrow></math></span>, in a linear ordering of their spectral radius when <span><math><mi>G</mi></math></span> is self-centered and <span><math><mi>H</mi></math></span> is any rooted graph whose root vertex has a fixed eccentricity.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 348-358"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144556704","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":"The average expected value of a rooted graph, Monte Carlo calculation, and a power index for the voting game","authors":"Masahiro Hachimori","doi":"10.1016/j.dam.2025.06.047","DOIUrl":"10.1016/j.dam.2025.06.047","url":null,"abstract":"<div><div>The expected value of a rooted graph is the expected value of the number of vertices reachable from the root when each edge is deleted independently with probability <span><math><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow></math></span>. The uniform expected value of a rooted graph is the average of the expected value assuming the probability <span><math><mi>p</mi></math></span> is taken uniformly at random from <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>. In this paper, after verifying that computing the uniform expected value of a rooted graph is #-P-hard, we propose a Monte Carlo method for computing the uniform expected value. We also discuss some applications of the proposed method for 0/1-valued monotone set functions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 74-86"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550027","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":"Submodular + Supermodular function maximization with knapsack constraint","authors":"Majun Shi ,&nbsp;Zishen Yang ,&nbsp;Wei Wang","doi":"10.1016/j.dam.2025.06.062","DOIUrl":"10.1016/j.dam.2025.06.062","url":null,"abstract":"<div><div>We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function <span><math><mi>f</mi></math></span> and a normalized monotone supermodular function <span><math><mi>g</mi></math></span> under a knapsack constraint. By utilizing the total curvature <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> of <span><math><mi>f</mi></math></span> and the supermodular curvature <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msup></math></span> of <span><math><mi>g</mi></math></span>, we demonstrate that this problem can achieve a near-optimal solution through three approaches: a greedy algorithm, an iterated submodular+modular procedure and a sandwich method. In particular, we prove that both the greedy algorithm and the iterated submodular+modular procedure provide an approximation guarantee of <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msup><mo>)</mo></mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></mrow></math></span>, while the sandwich method achieves a <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>e</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>-approximation ratio. All proposed algorithms run in polynomial time, and parameters such as <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> and <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msup></math></span> can be computed efficiently in linear time. Additionally, all three algorithms yield a <span><math><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msup><mo>)</mo></mrow></math></span>-approximation performance for knapsack-constrained monotone supermodular function maximization. Finally, we empirically test our first two algorithms on a constructed application. Although both algorithms have the same theoretical guarantee, their practical behavior differs significantly, leading to distinct solutions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 113-133"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550029","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":"Oriented Ramsey numbers of some sparse graphs","authors":"Junying Lu ,&nbsp;Yaojun Chen","doi":"10.1016/j.dam.2025.06.051","DOIUrl":"10.1016/j.dam.2025.06.051","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be an oriented graph without directed cycle. The oriented Ramsey number of <span><math><mi>H</mi></math></span>, denoted by <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, is the smallest integer <span><math><mi>N</mi></math></span> such that every tournament on <span><math><mi>N</mi></math></span> vertices contains a copy of <span><math><mi>H</mi></math></span>. Rosenfeld (JCT-B, 1974) conjectured that <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow></mrow></math></span> if <span><math><mi>H</mi></math></span> is a cycle of sufficiently large order, which was confirmed for <span><math><mrow><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow><mo>≥</mo><mn>9</mn></mrow></math></span> by Zein recently, and so does if <span><math><mi>H</mi></math></span> is a path. Note that <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow></mrow></math></span> implies any tournament contains <span><math><mi>H</mi></math></span> as a spanning subdigraph, it is interesting to ask when <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow></mrow></math></span> for <span><math><mi>H</mi></math></span> being a sparse oriented graph. Sós (1986) conjectured this is true if <span><math><mi>H</mi></math></span> is a directed path plus an additional edge containing the origin of the path as one end, which was confirmed by Petrović (JGT, 1988). In this paper, we show that <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow></mrow></math></span> for <span><math><mi>H</mi></math></span> being an oriented graph obtained by identifying a vertex of an antidirected cycle with one end of a directed path. Some other oriented Ramsey numbers for oriented graphs with one cycle are also discussed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 95-101"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557046","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":"Listing maximal H-free subgraphs","authors":"Alessio Conte ,&nbsp;Roberto Grossi ,&nbsp;Mamadou Moustapha Kanté ,&nbsp;Andrea Marino ,&nbsp;Takeaki Uno","doi":"10.1016/j.dam.2025.06.004","DOIUrl":"10.1016/j.dam.2025.06.004","url":null,"abstract":"<div><div>Given two graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, where <span><math><mi>H</mi></math></span> is the forbidden subgraph or pattern, <span><math><mi>G</mi></math></span> is called <span><math><mi>H</mi></math></span>-free if no vertex subset <span><math><mrow><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> induces a subgraph <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow></mrow></math></span> isomorphic to <span><math><mi>H</mi></math></span>. In the edge-induced version of the notion, <span><math><mi>G</mi></math></span> is called <span><math><mi>H</mi></math></span>-free if no edge subset <span><math><mrow><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊆</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> induces a subgraph <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow></mrow></math></span> isomorphic to <span><math><mi>H</mi></math></span>. The goal is to list all the inclusion-maximal subgraphs of <span><math><mi>G</mi></math></span> that are <span><math><mi>H</mi></math></span>-free, according to both the edge-induced and vertex-induced versions. Apart from its theoretical interest, the problem has application in data modeling, as it corresponds to data cleaning/repairing tasks, where the entire dataset is inconsistent with respect to the constraints given in <span><math><mi>H</mi></math></span>, and maximal consistent portions are sought. Several output-sensitive algorithms for the vertex-induced version are presented, which depend on the constraints on <span><math><mi>H</mi></math></span> and on <span><math><mi>G</mi></math></span>. As for the edge-induced version, we show how output-sensitive algorithms are possible for specific cases, but an efficient general technique is unlikely to exist as simply certifying a solution can be co-NP-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 18-31"},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534398","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}