{"title":"Bounds on independent isolation in graphs","authors":"Geoffrey Boyer, Wayne Goddard","doi":"10.1016/j.dam.2025.04.016","DOIUrl":"10.1016/j.dam.2025.04.016","url":null,"abstract":"<div><div>An isolating set of a graph is a set of vertices <span><math><mi>S</mi></math></span> such that, if <span><math><mi>S</mi></math></span> and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size of such a set. It is known that for every connected graph apart from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, the isolation number is at most one-third the order and indeed such a graph has three disjoint isolating sets. In this paper we consider isolating sets where <span><math><mi>S</mi></math></span> is required to be an independent set and call the minimum size thereof the independent isolation number. While for general graphs of order <span><math><mi>n</mi></math></span> the independent isolation number can be arbitrarily close to <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span>, we show that in bipartite graphs the vertex set can be partitioned into three disjoint independent isolating sets, whence the independent isolation number is at most <span><math><mrow><mi>n</mi><mo>/</mo><mn>3</mn></mrow></math></span>; while for 3-colorable graphs the maximum value of the independent isolation number is <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>3</mn></mrow></math></span>. We also provide a bound for <span><math><mi>k</mi></math></span>-colorable graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 143-149"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143850773","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":"Double Roman domination stability in graphs","authors":"Wei Zhuang","doi":"10.1016/j.dam.2025.04.035","DOIUrl":"10.1016/j.dam.2025.04.035","url":null,"abstract":"<div><div>A double Roman dominating function (DRDF) on a graph <span><math><mi>G</mi></math></span> is a function <span><math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow></mrow></math></span> having the property that if <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>, then the vertex <span><math><mi>v</mi></math></span> must have at least two neighbors assigned 2 under <span><math><mi>f</mi></math></span> or one neighbor <span><math><mi>w</mi></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span>, and if <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, then the vertex <span><math><mi>v</mi></math></span> must have at least one neighbor <span><math><mi>w</mi></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span>. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the minimum weight of a DRDF on <span><math><mi>G</mi></math></span>. The <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub></math></span><em>-stability</em> (<span><math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span><em>-stability</em>, <span><math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span><em>-stability</em>) of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>\u0000 (<span><math><mrow><mi>s</mi><msubsup><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub></mrow><mrow><mo>−</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>s</mi><msubsup><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the double Roman domination number. In this paper, we determine the exact values on the <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>d</mi><mi>R</mi></mrow></msub></math></span>-stability of some special classes of graphs, and present some bounds on <span><math><mrow><mi>s</mi><msub><mro","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 254-263"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847943","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":"Bounds on Graceful k-colorings of graphs","authors":"Paola T. Pantoja , Simone Dantas , Atílio G. Luiz","doi":"10.1016/j.dam.2025.04.032","DOIUrl":"10.1016/j.dam.2025.04.032","url":null,"abstract":"<div><div>We investigate the graceful <span><math><mi>k</mi></math></span>-coloring introduced by Gary Chartrand in 2015. The <em>graceful</em> <span><math><mi>k</mi></math></span><em>-coloring</em> of a graph <span><math><mi>G</mi></math></span> consists of a proper vertex coloring <span><math><mrow><mi>π</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</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>, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, that induces a proper edge coloring <span><math><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</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><mn>1</mn><mo>}</mo></mrow></mrow></math></span> defined by <span><math><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>π</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>π</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The smallest positive integer <span><math><mi>k</mi></math></span> for which a graph <span><math><mi>G</mi></math></span> has a graceful <span><math><mi>k</mi></math></span>-coloring is called the <em>graceful chromatic number</em> of <span><math><mi>G</mi></math></span> and is denoted by <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we improve a previous upper bound for the graceful chromatic number of an arbitrary graph, showing that <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></math></span>, where <span><math><mi>Δ</mi></math></span> is the maximum degree of the graph <span><math><mi>G</mi></math></span>. This also implies an improvement over the first bound given for complete graphs. Moreover, we study this problem within the context of cubic graph classes, determining the exact value of the graceful chromatic number of each member of the infinite families of the Generalized Blanuša and the Loupekine snarks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 218-231"},"PeriodicalIF":1.0,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847945","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":"Energy of a graph and Randić index of subgraphs","authors":"Gerardo Arizmendi, Diego Huerta","doi":"10.1016/j.dam.2025.04.029","DOIUrl":"10.1016/j.dam.2025.04.029","url":null,"abstract":"<div><div>We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mi>R</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the energy of a graph <span><math><mi>G</mi></math></span> and <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the Randić index of any subgraph of <span><math><mi>G</mi></math></span> (not necessarily induced). In particular, this generalizes well-known inequalities <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the matching number. We give other inequalities as applications to this result.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 136-142"},"PeriodicalIF":1.0,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843951","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":"Bicrucial k-power-free permutations","authors":"Margarita Akhmejanova , Aiya Kuchukova , Alexandr Valyuzhenich , Ilya Vorobyev","doi":"10.1016/j.dam.2025.04.030","DOIUrl":"10.1016/j.dam.2025.04.030","url":null,"abstract":"<div><div>In this work, we prove that for every <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span> there exist arbitrarily long bicrucial <span><math><mi>k</mi></math></span>-power-free permutations.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 232-239"},"PeriodicalIF":1.0,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843670","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 polynomial reconstruction problem for graphs having cut-vertices of degree two","authors":"Alexander Farrugia","doi":"10.1016/j.dam.2025.04.003","DOIUrl":"10.1016/j.dam.2025.04.003","url":null,"abstract":"<div><div>In the polynomial reconstruction problem (PRP), the characteristic polynomial <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> is sought from the polynomial deck <span><math><mrow><mi>PD</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> containing the characteristic polynomials of the <span><math><mi>n</mi></math></span> vertex-deleted subgraphs of <span><math><mi>G</mi></math></span>. The diagonal entries of the adjugate matrix <span><math><mrow><mo>adj</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> are the elements of <span><math><mrow><mi>PD</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The PRP is not completely solved for graphs having vertices of degree one. In this paper, we use <span><math><mrow><mo>adj</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> to successfully obtain <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> from <span><math><mrow><mi>PD</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for certain graphs having a vertex of degree one whose characteristic polynomial is not discoverable using results from current literature. Our methods require such graphs to have a vertex of degree one adjacent to a vertex of degree two, and this latter vertex would then be a cut-vertex of <span><math><mi>G</mi></math></span>. We thus extend this idea to partially solve the PRP for more general graphs that have a cut-vertex of degree two which is not necessarily adjacent to vertices of degree one, presenting an algorithm that provides <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> from <span><math><mrow><mi>PD</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> has such a cut-vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 165-175"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829891","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 average sizes and enumeration of minimal edge covers","authors":"John Engbers , Aysel Erey","doi":"10.1016/j.dam.2025.04.022","DOIUrl":"10.1016/j.dam.2025.04.022","url":null,"abstract":"<div><div>An <em>edge cover</em> <span><math><mi>M</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is a subset of edges such that every vertex of <span><math><mi>G</mi></math></span> is an end-vertex of some edge in <span><math><mi>M</mi></math></span>. An edge cover is called a <em>minimal edge cover</em> if it does not properly contain another edge cover. Let <span><math><mrow><mo>mec</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of minimal edge covers of <span><math><mi>G</mi></math></span>, and let <span><math><mrow><msub><mrow><mo>mec</mo></mrow><mrow><mi>av</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the average size of a minimal edge cover of <span><math><mi>G</mi></math></span>. We consider the extremal values of <span><math><mrow><mo>mec</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mo>mec</mo></mrow><mrow><mi>av</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is restricted to various families of graphs. In particular, we determine the graphs <span><math><mi>G</mi></math></span> which minimize <span><math><mrow><mo>mec</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among all graphs with fixed order, and among the family of 2-regular graphs with fixed order. We also determine the graphs which maximize <span><math><mrow><mo>mec</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> within the families of trees, of unicyclic graphs with fixed order, and of 2-regular graphs with a fixed order. Finally, we provide a characterization of all extremal graphs for the maximum and minimum values of <span><math><mrow><msub><mrow><mo>mec</mo></mrow><mrow><mi>av</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among all graphs <span><math><mi>G</mi></math></span> with fixed order.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 148-164"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829888","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}
Faisal N. Abu-Khzam , Emmanuel Arrighi , Matthias Bentert , Pål Grønås Drange , Judith Egan , Serge Gaspers , Alexis Shaw , Peter Shaw , Blair D. Sullivan , Petra Wolf
{"title":"Cluster Editing with Vertex Splitting","authors":"Faisal N. Abu-Khzam , Emmanuel Arrighi , Matthias Bentert , Pål Grønås Drange , Judith Egan , Serge Gaspers , Alexis Shaw , Peter Shaw , Blair D. Sullivan , Petra Wolf","doi":"10.1016/j.dam.2025.04.013","DOIUrl":"10.1016/j.dam.2025.04.013","url":null,"abstract":"<div><div><span>Cluster Editing</span>, also known as <span>Correlation Clustering</span>, is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to <span><math><mi>k</mi></math></span> edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. In this paper, we introduce a variation of <span>Cluster Editing</span> we call <span>Cluster Editing with Vertex Splitting</span> that extends this model to settings where clusters may be overlapping. Specifically, we allow a new edit operation that divides a vertex into two new vertices, each with a subset of the original neighbors. This approach addresses the limitations of assuming disjoint clusters, while still inherently limiting the amount of overlap when the number of edits is small. We show that <span>Cluster Editing with Vertex Splitting</span> is NP-complete and fixed-parameter tractable when parameterized by the number of editing operations <span><math><mi>k</mi></math></span>. In particular, we obtain <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>9</mn><mi>k</mi><mo>log</mo><mi>k</mi></mrow></msup><mo>+</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow></mrow></math></span>-time algorithm and a <span><math><mrow><mn>6</mn><mi>k</mi></mrow></math></span>-vertex kernel.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 185-195"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829798","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}
Júlio Araújo , Frédéric Havet , Claudia Linhares Sales , Nicolas Nisse , Karol Suchan
{"title":"Semi-proper orientations of dense graphs","authors":"Júlio Araújo , Frédéric Havet , Claudia Linhares Sales , Nicolas Nisse , Karol Suchan","doi":"10.1016/j.dam.2025.04.020","DOIUrl":"10.1016/j.dam.2025.04.020","url":null,"abstract":"<div><div>An <em>orientation</em> <span><math><mi>D</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is a digraph obtained from <span><math><mi>G</mi></math></span> by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation <span><math><mi>D</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is a <span><math><mi>k</mi></math></span><em>-orientation</em> if the in-degree of each vertex in <span><math><mi>D</mi></math></span> is at most <span><math><mi>k</mi></math></span>. An orientation <span><math><mi>D</mi></math></span> of <span><math><mi>G</mi></math></span> is <em>proper</em> if any two adjacent vertices have different in-degrees in <span><math><mi>D</mi></math></span>. The <em>proper orientation number</em> of a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mover><mrow><mi>χ</mi></mrow><mo>⃗</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum <span><math><mi>k</mi></math></span> such that <span><math><mi>G</mi></math></span> has a proper <span><math><mi>k</mi></math></span>-orientation.</div><div>A <em>weighted orientation</em> of a graph <span><math><mi>G</mi></math></span> is a pair <span><math><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>D</mi></math></span> is an orientation of <span><math><mi>G</mi></math></span> and <span><math><mi>w</mi></math></span> is an arc-weighting <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>→</mo><mi>N</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. A <em>semi-proper orientation</em> of <span><math><mi>G</mi></math></span> is a weighted orientation <span><math><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math></span> of <span><math><mi>G</mi></math></span> such that for every two adjacent vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> in <span><math><mi>G</mi></math></span>, we have that <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≠</mo><msub><mrow><mi>S</mi></mrow><mrow><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> is the sum of the weights of the arcs in <span><math><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math></span> with head <span><math><mi>v</mi></math></span>. For a positive integer <span><math><mi>k</mi></math></span>, a <em>semi-proper</em> <span><math><mi>k</mi></math></span><em>-orientation</em> <span><math><mrow><mo>(</mo><mi>D</mi><mo>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 196-217"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829890","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 results on injective edge coloring of graphs with maximum degree 5","authors":"Jian Lu , Xiang-Feng Pan","doi":"10.1016/j.dam.2025.04.021","DOIUrl":"10.1016/j.dam.2025.04.021","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is called injective <span><math><mi>k</mi></math></span>-edge colorable if it has a <span><math><mi>k</mi></math></span>-edge coloring <span><math><mi>φ</mi></math></span> such that any three consecutive edges <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>, and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> in the same path or triangle satisfy <span><math><mrow><mi>φ</mi><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>≠</mo><mi>φ</mi><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>. The injective chromatic index <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the smallest <span><math><mi>k</mi></math></span> such that <span><math><mi>G</mi></math></span> is injective <span><math><mi>k</mi></math></span>-edge colorable. We demonstrate that any graph with maximum degree 5 has <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> at most 7 (resp. 8, 9, 10) if its maximum average degree is less than <span><math><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> (resp. <span><math><mfrac><mrow><mn>12</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>, <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, <span><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>), which improves some results of Zhu <em>et al.</em> (2023) and Bu <em>et al.</em> (2024).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 176-184"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829889","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}
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