Academic Journal
On the total chromatic number of the direct product of cycles and complete graphs
| Τίτλος: | On the total chromatic number of the direct product of cycles and complete graphs |
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| Συγγραφείς: | Diane Castonguay, Celina M. H. de Figueiredo, Luis Antonio Brasil Kowada, Caroline Patrão, Diana Sasaki, Mario Valencia-Pabon |
| Πηγή: | RAIRO - Operations Research. 58:1609-1632 |
| Στοιχεία εκδότη: | EDP Sciences, 2024. |
| Έτος έκδοσης: | 2024 |
| Θεματικοί όροι: | Product (mathematics), 4. Education, Geometry, 0102 computer and information sciences, Graph Labeling, 01 natural sciences, Computer science, Optical Code Division Multiple Access, Engineering, Computational Theory and Mathematics, Combinatorics, Computer Science, Physical Sciences, Chromatic scale, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, 0101 mathematics, Electrical and Electronic Engineering, Graph Labeling and Dimension Problems, Mathematics, Graph Theory and Algorithms |
| Περιγραφή: | Ak-total coloringof a graphGis an assignment ofkcolors to the elements (vertices and edges) ofGso that adjacent or incident elements have different colors. The total chromatic number is the smallest integerkfor whichGhas ak-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Δ(G) + 1 (called Type 1) or Δ(G) + 2 (called Type 2), where Δ(G) is the maximum degree ofG. We consider the direct product of complete graphsKm×Kn. It is known that if at least one of the numbersmornis even, thenKm×Knis Type 1, except forK2×K2. We prove that the graphKm×Knis Type 1 when bothmandnare odd numbers, by using that the conformable condition is sufficient for the graphKm×Knto be Type 1 when bothmandnare large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct productCm×Knof a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite familyCm×Kn, whenmis not a multiple of 3 andn= 2. We provide evidence to conjecture that all otherCm×Knare Type 1. |
| Τύπος εγγράφου: | Article Other literature type |
| ISSN: | 2804-7303 0399-0559 |
| DOI: | 10.1051/ro/2024045 |
| DOI: | 10.60692/v0ajp-hkw64 |
| DOI: | 10.60692/w9hdx-j8694 |
| Rights: | CC BY |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....acf242f8e6d19e65d87e1a26d9c539d5 |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 28047303 03990559 |
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| DOI: | 10.1051/ro/2024045 |