Anna Flaszczyńska, Aleksandra Gorzkowska, Mariusz Woźniak
{"title":"A note on sequences variant of irregularity strength for hypercubes","authors":"Anna Flaszczyńska, Aleksandra Gorzkowska, Mariusz Woźniak","doi":"10.1016/j.amc.2025.129312","DOIUrl":null,"url":null,"abstract":"Let <mml:math altimg=\"si1.svg\"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:math> be an edge-coloring of the <ce:italic>n</ce:italic>-dimension hypercube <mml:math altimg=\"si2.svg\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. By the palette at a vertex <ce:italic>v</ce:italic> we mean the sequence <mml:math altimg=\"si3.svg\"><mml:mo stretchy=\"true\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"true\">)</mml:mo></mml:math>, where <mml:math altimg=\"si4.svg\"><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> is the edge incident to <ce:italic>v</ce:italic> that connects vertices differing in the <ce:italic>i</ce:italic>th element. In this paper, we show that two colors are enough to distinguish all vertices of the <ce:italic>n</ce:italic>-dimensional hypercube <mml:math altimg=\"si2.svg\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math altimg=\"si18.svg\"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math>) by their palettes. We also show that if <ce:italic>f</ce:italic> is a proper edge-coloring of the hypercube <mml:math altimg=\"si2.svg\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math altimg=\"si6.svg\"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math>), then <ce:italic>n</ce:italic> colors suffice to distinguish all vertices by their palettes.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"58 1 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.amc.2025.129312","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let f:E→{1,2,…,k} be an edge-coloring of the n-dimension hypercube Hn. By the palette at a vertex v we mean the sequence (f(e1(v)),f(e1(v)),…,f(en(v))), where ei(v) is the edge incident to v that connects vertices differing in the ith element. In this paper, we show that two colors are enough to distinguish all vertices of the n-dimensional hypercube Hn (n≥2) by their palettes. We also show that if f is a proper edge-coloring of the hypercube Hn (n≥5), then n colors suffice to distinguish all vertices by their palettes.
期刊介绍：
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.
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