Discrete Mathematics最新文献

{"title":"Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy","authors":"Masaki Kashima ,&nbsp;Riste Škrekovski ,&nbsp;Rongxing Xu","doi":"10.1016/j.disc.2026.115003","DOIUrl":"10.1016/j.disc.2026.115003","url":null,"abstract":"<div><div>A proper coloring <em>ϕ</em> of <em>G</em> is called a proper conflict-free coloring of <em>G</em> if for every non-isolated vertex <em>v</em> of <em>G</em>, there is a color <em>c</em> such that <span><math><mo>|</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>∩</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosability of graphs. For a non-negative integer <em>k</em>, a graph <em>G</em> is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosable if for any list assignment <em>L</em> of <em>G</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>+</mo><mi>k</mi></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>G</em> admits a proper conflict-free coloring <em>ϕ</em> such that <span><math><mi>ϕ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>∈</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this note, we first remark if a graph <em>G</em> is <em>d</em>-degenerate, then <em>G</em> is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable. Furthermore, when <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, we can reduce the number of colors by showing that every tree is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable. This motivates us to state a question.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115003"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079006","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":"Perfect proper edge colorings of regular bipartite graphs with rainbow C4-s","authors":"András Gyárfás ,&nbsp;Gábor N. Sárközy ,&nbsp;Adam Zsolt Wagner","doi":"10.1016/j.disc.2026.115005","DOIUrl":"10.1016/j.disc.2026.115005","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We call a proper edge coloring of a bipartite graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; a B-coloring if every 4-cycle of &lt;em&gt;G&lt;/em&gt; is colored with four different colors. Denote by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; the smallest number of colors needed for a B-coloring of graph &lt;em&gt;G&lt;/em&gt;. The question whether &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; implies &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is from Burr, Erdős, Graham and Sós. A positive answer to this question would imply a positive answer to the famous &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-conjecture of Brown, Erdős and Sós. Here we look at an interesting test case of this question. We call a B-coloring of a &lt;em&gt;d&lt;/em&gt;-regular bipartite graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;em&gt;perfect&lt;/em&gt; if each color class forms a perfect matching in &lt;em&gt;G&lt;/em&gt; (i.e. has &lt;em&gt;n&lt;/em&gt; edges).&lt;/div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the minimal &lt;em&gt;n&lt;/em&gt; such that there exists a perfect B-coloring of some &lt;em&gt;d&lt;/em&gt;-regular bipartite graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A test case of the question above is whether &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is super-linear. We prove the affirmative answer for &lt;em&gt;shifted colorings&lt;/em&gt;: defined by the matchings &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115005"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980014","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":"Density of 2-critical signed graphs","authors":"Reza Naserasr ,&nbsp;Weiqiang Yu","doi":"10.1016/j.disc.2026.114998","DOIUrl":"10.1016/j.disc.2026.114998","url":null,"abstract":"<div><div>A balanced <em>k</em>-coloring of a signed graph <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> which has no negative loop is to partition its vertices into <em>k</em> sets each of which induces a balanced subgraph, that is a subgraph with no negative cycle. The notion, through basic graph operation, captures the classic proper coloring of graphs as special case.</div><div>Having observed the importance of balanced 2-coloring, in this work we study structural conditions which permit a signed graph to admit a 2-coloring. More precisely, slightly modifying the notion of color-critical, we say a signed graph is <em>k</em>-critical if it admits no balanced <em>k</em>-coloring but every proper subgraph of it admits such a coloring.</div><div>We show that if <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is a 2-critical signed graph whose underlying graph is not <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> or an odd cycle, then<span><span><span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mfrac><mrow><mn>21</mn><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></mfrac></math></span></span></span> where <em>d</em> is the maximum number of vertex disjoint digons in <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>. As a corollary we conclude that, except for the signed graph <span><math><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mo>−</mo><mo>)</mo></math></span>, any signed simple graph with maximum average degree at most 4.2 admits a balanced 2-coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114998"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980076","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":"Structure and linear-Pollyanna for some square-free graphs","authors":"Ran Chen,&nbsp;Baogang Xu","doi":"10.1016/j.disc.2026.114979","DOIUrl":"10.1016/j.disc.2026.114979","url":null,"abstract":"<div><div>We use <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> to denote a path and a cycle on <em>t</em> vertices, respectively. A <em>bull</em> is a graph consisting of a triangle with two disjoint pendant edges, a <em>hammer</em> is a graph obtained by identifying an endvertex of a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> with a vertex of a triangle. A class <span><math><mi>F</mi></math></span> is <em>χ</em>-bounded if there is a function <em>f</em> such that <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> for all induced subgraphs <em>G</em> of a graph in <span><math><mi>F</mi></math></span>. A class <span><math><mi>C</mi></math></span> of graphs is <em>Pollyanna</em> (resp. <em>linear-Pollyanna</em>) if <span><math><mi>C</mi><mo>∩</mo><mi>F</mi></math></span> is polynomially (resp. linearly) <em>χ</em>-bounded for every <em>χ</em>-bounded class <span><math><mi>F</mi></math></span> of graphs. Chudnovsky et al. <span><span>[6]</span></span> showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let <em>G</em> be a connected graph with no clique cutsets and no universal vertices. In this paper, we show that <em>G</em> is <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, hammer)-free if and only if it has girth at least 5, and <em>G</em> is <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, bull)-free graphs and <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, hammer)-free graphs are linear-Pollyannas. We also show that the class of (bull, diamond)-free graphs is linear-Pollyanna.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114979"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928976","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 non-traceable 3-connected planar cubic graphs of minimum order","authors":"Gholamreza Abrishami ,&nbsp;Freydoon Rahbarnia ,&nbsp;Nico Van Cleemput","doi":"10.1016/j.disc.2026.114983","DOIUrl":"10.1016/j.disc.2026.114983","url":null,"abstract":"<div><div>In 1980, Zamfirescu presented a non-traceable (i.e. without a hamiltonian path) 3-connected planar cubic graph with 88 vertices, which is still the smallest known one of its kind. In this note we present several new examples with 88 vertices. Moreover, we present a non-traceable cyclically 4-connected planar cubic graph with 168 vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114983"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980071","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":"Solutions to open problems on the exponential augmented Zagreb index","authors":"Kinkar Chandra Das ,&nbsp;Da-yeon Huh ,&nbsp;Sourav Mondal","doi":"10.1016/j.disc.2025.114967","DOIUrl":"10.1016/j.disc.2025.114967","url":null,"abstract":"<div><div>The exponential augmented Zagreb (<em>EAZ</em>) index is a graph-theoretical descriptor that correlates strongly with the physico-chemical properties of molecules. Introduced by Rada in 2019, it is defined for a simple graph ϒ as<span><span><span><math><mi>E</mi><mi>A</mi><mi>Z</mi><mo>(</mo><mi>ϒ</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>ϒ</mi><mo>)</mo></mrow></munder><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>E</mi><mo>(</mo><mi>ϒ</mi><mo>)</mo></math></span> denotes the edge set and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the degree of vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. This work is motivated by some open problems concerning the well-known augmented Zagreb index (<em>AZ</em>). In particular, the maximization of <em>AZ</em> for a given graph order and a specified number of pendant vertices was posed as an open problem in Chen et al. (2022) <span><span>[7]</span></span>. We completely resolve this problem for the exponential version, <em>EAZ</em>. In recent work Xu et al. (2025) <span><span>[38]</span></span>, two related questions were raised: whether the maximal graphs for <em>AZ</em> and <em>EAZ</em> coincide, and if not, how they differ. We provide complete answers to these questions with respect to the chromatic number and the number of pendant vertices. We explore the maximal graph for <em>EAZ</em> in terms of chromatic number and graph order, and show that this differs substantially from the corresponding extremal graph for <em>AZ</em>. Further results include a characterization of the maximal graphs for <em>EAZ</em> when vertex connectivity and edge connectivity are prescribed together with the graph order. In addition, we prove that <span><math><mi>E</mi><mi>A</mi><mi>Z</mi><mo>(</mo><mi>ϒ</mi><mo>)</mo></math></span> increases upon adding an edge to ϒ, a crucial result for understanding the extremal properties of <em>EAZ</em>. Finally, the potential usefulness of this discrete invariant in chemical graph theory is demonstrated.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114967"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928975","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":"Deletable edges in 3-connected graphs and their applications","authors":"S.R. Kingan","doi":"10.1016/j.disc.2026.114994","DOIUrl":"10.1016/j.disc.2026.114994","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt; be simple 3-connected graphs such that &lt;em&gt;G&lt;/em&gt; has an &lt;em&gt;H&lt;/em&gt;-minor. An edge &lt;em&gt;e&lt;/em&gt; in &lt;em&gt;G&lt;/em&gt; is called &lt;em&gt;H-deletable&lt;/em&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is 3-connected and has an &lt;em&gt;H&lt;/em&gt;-minor. The main result in this paper establishes that, if &lt;em&gt;G&lt;/em&gt; has no &lt;em&gt;H&lt;/em&gt;-deletable edges, then there exists a sequence of simple 3-connected graphs &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with no &lt;em&gt;H&lt;/em&gt;-deletable edges such that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; one of three possibilities holds: &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;; &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; where &lt;em&gt;e&lt;/em&gt; and &lt;em&gt;f&lt;/em&gt; are incident to a degree 3 vertex in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;; or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; where &lt;em&gt;w&lt;/em&gt; is a degree 3 vertex in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. We give several applications including a graph-theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph. Halin proved that if &lt;em&gt;G&lt;/em&gt; is a minimally 3-connected graph on &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; vertices, then &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and equality holds if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. We give a different proof of Halin's result and extend it by identifying the minimally 3-connected infinite family of graphs with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Finally, we extend the main theorem to mat","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114994"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038450","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":"Improved bounds for proper rainbow saturation","authors":"Andrew Lane ,&nbsp;Natasha Morrison","doi":"10.1016/j.disc.2026.115001","DOIUrl":"10.1016/j.disc.2026.115001","url":null,"abstract":"<div><div>Given a graph <em>H</em>, we say that a graph <em>G</em> is <em>properly rainbow H-saturated</em> if: (1) There is a proper edge colouring of <em>G</em> containing no rainbow copy of <em>H</em>; (2) For every <span><math><mi>e</mi><mo>∉</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, every proper edge colouring of <span><math><mi>G</mi><mo>+</mo><mi>e</mi></math></span> contains a rainbow copy of <em>H</em>. The <em>proper rainbow saturation number</em> <span><math><mrow><mi>sa</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the minimum number of edges in a properly rainbow <em>H</em>-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on <span><math><mrow><mi>sa</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for general cliques, cycles, and complete bipartite graphs. We also provide a necessary and sufficient condition for a graph to have constant proper rainbow saturation number.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115001"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079004","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 Kneser chromatic function distinguishes trees","authors":"Yusaku Nishimura","doi":"10.1016/j.disc.2026.115009","DOIUrl":"10.1016/j.disc.2026.115009","url":null,"abstract":"<div><div>R.P. Stanley defined an invariant for graphs called the chromatic symmetric function and conjectured it is a complete invariant for trees. Miezaki et al. generalized the chromatic symmetric function and defined the Kneser chromatic functions denoted by <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></msub><mo>}</mo></mrow><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, and rephrase Stanley's conjecture that <span><math><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi><mo>,</mo><mn>1</mn></mrow></msub></mrow></msub></math></span> is a complete invariant for trees. This paper shows <span><math><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi><mo>,</mo><mn>2</mn></mrow></msub></mrow></msub></math></span> is a complete invariant for trees.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115009"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038446","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":"Quasi-cyclic codes of index 2","authors":"Kanat Abdukhalikov ,&nbsp;Askar S. Dzhumadil'daev ,&nbsp;San Ling","doi":"10.1016/j.disc.2026.115004","DOIUrl":"10.1016/j.disc.2026.115004","url":null,"abstract":"<div><div>We study quasi-cyclic codes of index 2 over finite fields. We give a classification of such codes. Their duals with respect to the Euclidean, symplectic and Hermitian inner products are investigated. We describe self-orthogonal and dual-containing codes. Lower bounds for minimum distances of quasi-cyclic codes are given. A quasi-cyclic code of index 2 is generated by at most two elements. We describe conditions when such a code (or its dual) is generated by one element.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115004"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038449","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}