| Title: | On k-rainbow total domination and a related conjecture |
|---|
| Authors: | ID Erveš, Rija (Author)
ID Kraner Šumenjak, Tadeja (Author)
ID Tepeh, Aleksandra (Author) |
|---|
| Files: |  s40840-026-02060-2.pdf (367,71 KB)
MD5: AB2F763022F9F533F7A0D78D75866762
|
|---|
| Language: | English |
|---|
| Work type: | Article |
|---|
| Typology: | 1.01 - Original Scientific Article |
|---|
| Organization: | FGPA - Faculty of Civil Engineering, Transportation Engineering and Architecture
FERI - Faculty of Electrical Engineering and Computer Science
FKBV - Faculty of Agriculture and Life Sciences
|
|---|
| Abstract: | A k-rainbow total dominating function of a graph G is a function f : V(G) → 2[k] such that for every vertex v ∈ V(G) with f (v) = ∅, the union of the colors assigned to its neighbors equals [k], and if f (v) = {i}, then v has a neighbor u with i ∈ f (u). The minimum weight of such a function is called the k-rainbow total domination number of G and is denoted by γkrt(G). We contribute to the study of k-rainbow total domination by proving one conjecture and constructing a counterexample to another. First, we show that the problem of determining whether a graph admits a k-rainbow total dominating function of a given weight is NP-complete. In the second part, we derive an upper bound on the domination number of a graph G in terms of γkrt(G) and the frequency of the least-used color in a k-rainbow total dominating function. This result not only provides an alternative and shorter proof of a known lower bound on γkrt(G)/γ (G), originally established by Ojakian et al. (2021), but also contributes to disproving their conjecture on the lower bound for γkrt(G) when k = 4. |
|---|
| Keywords: | dominaiton, rainbow total domination, NP-complete |
|---|
| Publication status: | Published |
|---|
| Publication version: | Version of Record |
|---|
| Submitted for review: | 23.06.2025 |
|---|
| Article acceptance date: | 03.02.2026 |
|---|
| Publication date: | 19.02.2026 |
|---|
| Publisher: | Springer |
|---|
| Year of publishing: | 2026 |
|---|
| Number of pages: | 114 str. |
|---|
| Numbering: | Vol. 49, [article no.] 62 |
|---|
| PID: | 20.500.12556/DKUM-97220  |
|---|
| UDC: | 51 |
|---|
| ISSN on article: | 2180-4206 |
|---|
| COBISS.SI-ID: | 269154307 |
|---|
| DOI: | 10.1007/s40840-026-02060-2 |
|---|
| Copyright: | © The Author(s) 2026 |
|---|
| Publication date in DKUM: | 24.02.2026 |
|---|
| Views: | 99 |
|---|
| Downloads: | 7 |
|---|
| Metadata: |    |
|---|
| Categories: | Misc.
|
|---|
|
:
|
Copy citation |
|---|
| | | | Average score: | (0 votes) |
|---|
| Your score: | Voting is allowed only for logged in users. |
|---|
| Share: |  |
|---|
Hover the mouse pointer over a document title to show the abstract or click
on the title to get all document metadata. |
