Discrete Applied Mathematics最新文献

{"title":"Three results towards approximation of special maximum matchings in graphs","authors":"Vahan Mkrtchyan","doi":"10.1016/j.dam.2025.04.005","DOIUrl":"10.1016/j.dam.2025.04.005","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span> define the parameters <span><math><mrow><mi>ℓ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> as the minimum and maximum value of <span><math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>∖</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>F</mi></math></span> is a maximum matching of <span><math><mi>G</mi></math></span> and <span><math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the matching number of <span><math><mi>G</mi></math></span>. In this paper, we show that there is a small constant <span><math><mrow><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, such that the following decision problem is NP-complete: given a graph <span><math><mi>G</mi></math></span> and <span><math><mrow><mi>k</mi><mo>≤</mo><mfrac><mrow><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, check whether there is a maximum matching <span><math><mi>F</mi></math></span> in <span><math><mi>G</mi></math></span>, such that <span><math><mrow><mrow><mo>|</mo><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>∖</mo><mi>F</mi><mo>)</mo></mrow><mo>−</mo><mi>k</mi><mo>|</mo></mrow><mo>≤</mo><mi>c</mi><mi>⋅</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow></math></span>. Note that when <span><math><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow></math></span>, this problem is polynomial time solvable as we observe in the paper. Since in any graph <span><math><mi>G</mi></math></span>, we have <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>ℓ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, any polynomial time algorithm constructing a maximum matching of a graph is a 2-approximation algorithm for <span><math><mrow><mi>ℓ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>-approximation algorithm for <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We complement these observations by presenting two inapproximability results for <span><math><mrow><mi>ℓ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 127-136"},"PeriodicalIF":1.0,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143823882","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":"Forbidden Pairs of disconnected graphs for traceability and hamiltonicity","authors":"Hongli Liao,&nbsp;Qiang Wang,&nbsp;Liming Xiong,&nbsp;Zhang Zhang","doi":"10.1016/j.dam.2025.04.002","DOIUrl":"10.1016/j.dam.2025.04.002","url":null,"abstract":"<div><div>In this paper, we firstly characterize all forbidden pairs <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span> for graphs with a spanning trail that are traceable and traceable graphs that are hamiltonian. There is no change of forbidden pairs for hamiltonicity if we impose a necessary condition of assumption that the graph is traceable; however, there is some difference of forbidden pairs for traceability if we impose a necessary condition that the graph has a spanning trail: different on two pairs of forbidden subgraphs <span><math><mrow><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></mrow><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow></math></span> (where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the graph obtained by identifying a vertex of a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> with an end-vertex of a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>).</div><div>As a byproduct, we prove that if <span><math><mi>G</mi></math></span> is a connected <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></mrow></math></span>-free graph, then every subgraph <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>T</mi><mo>]</mo></mrow></mrow></math></span> induced by a trail <span><math><mi>T</mi></math></span> is traceable and every subgraph <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>T</mi><mo>]</mo></mrow></mrow></math></span> induced by a closed trail <span><math><mi>T</mi></math></span> is either hamiltonian or <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∨</mo><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143823402","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":"Finding spanning trees with perfect matchings","authors":"Kristóf Bérczi ,&nbsp;Tamás Király ,&nbsp;Yusuke Kobayashi ,&nbsp;Yutaro Yamaguchi ,&nbsp;Yu Yokoi","doi":"10.1016/j.dam.2025.04.001","DOIUrl":"10.1016/j.dam.2025.04.001","url":null,"abstract":"<div><div>We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the positive side, we design a simple greedy algorithm for the case when the graph is complete (or complete bipartite) and the edge weights take at most two values. On the negative side, the problem is NP-hard even when the graph is complete (or complete bipartite) and the edge weights take at most three values, or when the graph is cubic, planar, and bipartite and the edge weights take at most two values.</div><div>We also consider an interesting variant. We call a tree strongly balanced if on one side of the bipartition of the vertex set with respect to the tree, all but one of the vertices have degree 2 and the remaining one is a leaf. This property is a sufficient condition for a tree to have a perfect matching, which enjoys some structural property in addition. When the underlying graph is bipartite, strongly balanced spanning trees can be written as matroid intersection, and this fact was utilized to design approximation algorithms for several NP-hard problems, e.g., the traveling salesman problem and a kind of connectivity augmentation problem. The natural question is its tractability in nonbipartite graphs. As a negative answer, it turns out NP-hard to test whether a given graph has a strongly balanced spanning tree or not even when the graph is subcubic and planar.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 137-147"},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815362","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":"Fault-tolerant strong Menger connectivity with conditional faults on wheel networks","authors":"Shiying Wang ,&nbsp;Lina Zhao ,&nbsp;Su Wang","doi":"10.1016/j.dam.2025.04.019","DOIUrl":"10.1016/j.dam.2025.04.019","url":null,"abstract":"&lt;div&gt;&lt;div&gt;An interconnection network is usually modeled as a graph, where vertex and edge correspond the processor and the link between two distinct processors, respectively. Connectivity is an important metric for the fault tolerance in interconnection networks. A connected graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is called strongly Menger connected if each pair of vertices &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are connected by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;min&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; vertex-disjoint paths in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is called &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-fault-tolerant strongly Menger (&lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-FTSM for short) connected if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; remains strongly Menger connected for any &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. A graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is called &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-conditional fault-tolerant strongly Menger (&lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-CFTSM for short) connected if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; remains strongly Menger connected for any &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we show that &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-dimensional wheel network &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-FTSM connected for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-fault-tolerant one-to-many strongly Menger connected for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-CFTSM connected for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, the bounds &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 115-126"},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821098","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 cyclic diagnosability of Cayley graphs generated by transposition trees","authors":"Weixing Zheng ,&nbsp;Shuming Zhou ,&nbsp;Eddie Cheng ,&nbsp;Qifan Zhang","doi":"10.1016/j.dam.2025.04.012","DOIUrl":"10.1016/j.dam.2025.04.012","url":null,"abstract":"<div><div>The system’s self diagnosis capability is a critical indicator for the reliability of multiprocessor systems, which plays a key role in ensuring the smooth operation of big data processing and cloud computing. Cycle structures, which bring redundant paths for secure communication, enhance network reliability significantly. Based on cyclic connectivity, cyclic diagnosability has been proposed and has witnessed a small amount of progress. Although the metric relationship between cyclic connectivity and cyclic diagnosability has been established, the flexibility of generating graph of Cayley graphs prevents this metric relationship from being directly applied to characterize the cyclic diagnosability of all Cayley graphs generated by transposition trees. In this paper, we classify the generating transposition trees based on the diameter and the number of matchings, and determine the cyclic diagnosability of Cayley graphs generated by these transposition trees under PMC and MM* models.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 109-123"},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817412","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":"Distance-regular Cayley graphs over (pseudo-) semi-dihedral groups","authors":"Xueyi Huang ,&nbsp;Lu Lu ,&nbsp;Xiongfeng Zhan","doi":"10.1016/j.dam.2025.04.006","DOIUrl":"10.1016/j.dam.2025.04.006","url":null,"abstract":"<div><div>Distance-regular graphs constitute an important class of regular graphs exhibiting remarkable combinatorial symmetry. In 2007, Miklavič and Potočnik initiated the systematic study of distance-regular Cayley graphs, extending the classical problem of characterizing strongly regular Cayley graphs (equivalent to investigating regular partial difference sets). In this paper, we provide a partial characterization of distance-regular Cayley graphs over semi-dihedral groups and pseudo-semi-dihedral groups, both of which are 2-groups that contain a cyclic subgroup of index 2.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 95-108"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817749","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 note on extremal trees for a bound on the double domination number","authors":"Ravi Kalaiyarasi ,&nbsp;Mustapha Chellali ,&nbsp;Yanamandram B. Venkatakrishnan","doi":"10.1016/j.dam.2025.04.007","DOIUrl":"10.1016/j.dam.2025.04.007","url":null,"abstract":"<div><div>In a graph <span><math><mrow><mi>G</mi><mo>,</mo></mrow></math></span> a vertex is said to dominate itself and its neighbors. A subset <span><math><mi>D</mi></math></span> of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is a double dominating set if every vertex of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is dominated at least twice by the vertices of <span><math><mi>D</mi></math></span>. The double domination number <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mo>×</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the minimum cardinality among all double dominating sets of <span><math><mi>G</mi></math></span>. Cabrera-Martínez proved that for every nontrivial tree <span><math><mi>T</mi></math></span> of order <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>ℓ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> leaves and <span><math><mrow><mi>s</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> support vertices, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mo>×</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>n</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>−</mo><mi>γ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mi>ℓ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mi>s</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> stands for the domination number of <span><math><mrow><mi>T</mi><mo>.</mo></mrow></math></span> In this note, we provide a constructive characterization of trees attaining this bound in response to the problem raised by Cabrera-Martínez.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 71-75"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808184","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 the outdegree zeroth-order general Randić index of digraphs","authors":"Jiaxiang Yang ,&nbsp;Hechao Liu ,&nbsp;Xuesong Fu","doi":"10.1016/j.dam.2025.04.010","DOIUrl":"10.1016/j.dam.2025.04.010","url":null,"abstract":"<div><div>For any digraph <span><math><mi>D</mi></math></span> and real number <span><math><mi>α</mi></math></span>, the outdegree zeroth-order general Randić index is defined as the sum of the outdegree of each vertex raised to the power of <span><math><mi>α</mi></math></span> across all vertices in the digraph <span><math><mi>D</mi></math></span>. A cactus graph <span><math><mi>G</mi></math></span> is a connected graph in which each block of <span><math><mi>G</mi></math></span> is either an edge or a cycle. An oriented cactus is the directed variant of a cactus graph, where each edge has a specified direction. For any real number <span><math><mrow><mi>α</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and positive integers <span><math><mi>n</mi></math></span>, <span><math><mi>r</mi></math></span> with <span><math><mrow><mi>n</mi><mo>&gt;</mo><mn>2</mn><mi>r</mi></mrow></math></span>, we address the problem of identifying the maximum value of the outdegree zeroth-order general Randić index among oriented cacti with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>r</mi></math></span> cycles. Additionally, we determine the maximum value of the outdegree zeroth-order general Randić index over connected simple digraphs with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>l</mi></math></span> arcs, where <span><math><mi>l</mi></math></span> is a positive integer. In particular, for <span><math><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></math></span>, we obtain some of the results obtained in Ganie and Pirzada, (2024).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 76-86"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817748","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":"Rainbow transitive triangles in arc-colored digraphs","authors":"Mengyu Duan ,&nbsp;Zhiwei Guo ,&nbsp;Binlong Li ,&nbsp;Shenggui Zhang","doi":"10.1016/j.dam.2025.04.011","DOIUrl":"10.1016/j.dam.2025.04.011","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A subdigraph of an arc-colored digraph is &lt;em&gt;rainbow&lt;/em&gt; if its all arcs have distinct colors. For two digraphs &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the minimum integer such that every arc-colored digraph &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mrow&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;mi&gt;r&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; contains a rainbow copy of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the number of colors of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;↔&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; be the digraph obtained from the complete graph &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; by replacing each edge &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with a pair of symmetric arcs &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;⃗&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; be the transitive triangle. In this paper we determine &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;↔&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;⃗&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and characterize the corresponding extremal arc-colorings of &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;↔&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;. Further, we prove that an arc-colored digraph &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; on &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; vertices contains a rainbow &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;⃗&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow&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;mi&gt;c&lt;/mi&gt;&lt;mrow&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;mi&gt;a&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;↔&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;↔&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 175-184"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807898","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":"Unitary Cayley graphs of finite semisimple rings","authors":"Thanatach Tangsakul,&nbsp;Yotsanan Meemark","doi":"10.1016/j.dam.2025.04.008","DOIUrl":"10.1016/j.dam.2025.04.008","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a finite ring with identity &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;em&gt;unitary Cayley graph&lt;/em&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, denoted by &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;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, is a graph whose vertices are elements of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; are adjacent if and only if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a unit in &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A ring &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is called a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-ring if for any ring &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&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;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we study unitary Cayley graphs of finite semisimple rings and show that finite semisimple rings of the form &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; are &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-rings. We achieve this by using the Cartesian skeleton of the graph to show 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;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is prime with respect to the direct product and deduce the result through the uniqueness of graph factorization. In addition, we show that any finite commutative reduced ring is a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-ring by combining graph cancellation properties with combinatorial arguments on the eigenvalues of the unitary Cayley graph o","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 87-94"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817411","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}