| Title: | On the $\Delta$-edge stability number of graphs |
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| Authors: | ID Akbari, Saieed (Author)
ID Hosseini Dolatabadi, Reza (Author)
ID Jamaali, Mohsen (Author)
ID Klavžar, Sandi (Author)
ID Movarraei, Nazanin (Author) |
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| Files: |  PDF - Presentation file, download (611,25 KB)
MD5: 0283BCF233BF6C5CFE16A9A51281BF8C
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0195669825000502
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | The $\Delta$-edge stability number ${\rm es}_{\Delta}(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $\Delta(H) = \Delta(G)-1$. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order $2$ or $3$. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form ${\rm es}_{\Delta}(G) \leq c |V(G)|$ for an arbitrary graph $G$ of given maximum degree $\Delta$, where $c$ is a given constant, it suffices to prove the bound for $\Delta$-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if $\Delta(G) \geq\frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $\Delta(G)$-edge coloring, then ${\rm es}_{\Delta}(G) \le {\lceil |V(G)|/2\rceil}$. |
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| Keywords: | vertex degree, ▫$\Delta$▫-edge stability number, matching, edge coloring |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.06.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 10 str. |
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| Numbering: | Vol. 127, [article no.] 104167 |
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| PID: | 20.500.12556/DiRROS-22176  |
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| UDC: | 519.17 |
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| ISSN on article: | 0195-6698 |
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| DOI: | 10.1016/j.ejc.2025.104167 |
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| COBISS.SI-ID: | 234762499 |
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| Note: |
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| Publication date in DiRROS: | 07.05.2025 |
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| Views: | 534 |
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| Downloads: | 248 |
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