| Title: | A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs |
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| Authors: | ID Abreu, Marién (Author)
ID Baptist Gauci, John (Author)
ID Mattiolo, Davide (Author)
ID Mazzuoccolo, Giuseppe (Author)
ID Romaniello, Federico (Author)
ID Rubio-Montiel, Christian (Author)
ID Traetta, Tommaso (Author) |
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| Files: |  ADAM_Abreu,Baptist_Gauci,Mattiolo,Mazzuoccolo,Romaniello,Rubio-Montiel,Traetta_2025.pdf (387,22 KB)
MD5: C496415C3AEC0C98F4303E36045C0337
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| Abstract: | A proper total colouring of a graph G is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edgeswith their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of G, denoted by h_t(G). Here, we give a general upper bound for h_t(G) in terms of the order n of G. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph Kn and of the complete multigraph λK_n, where λ is the number of edges joining each pair of vertices of Kn. In particular, Araujo-Pardo et al. have recently shown that 3/2 n ≤ h_t(K_n)≤ 5/3 n + θ(1). In this paper, we prove that h_t(K_n) = ⌈3/2 n⌉ except for h_t(K₁) = 1 and h_t(K₄) = 7; therefore, h_t(G)≤ ⌈3/2 n⌉, for every graph G on n > 4 vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph λKn and as a consequence show that h_t(G) ≤ (λ-1)(2⌈n/2⌉-1)+⌈3n/2⌉ for n > 4, where G is a multigraph such that λ is the maximum number of edges between any two vertices. |
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| Keywords: | total colouring, harmonious colouring, complete graphs, complete multigraphs, Levi graph |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 12.12.2024 |
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| Publisher: | Založba Univerze na Primorskem |
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| Year of publishing: | 2025 |
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| Number of pages: | 10 str. |
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| Numbering: | Vol 8, no. 3, [article no.] P3.02 |
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| PID: | 20.500.12556/RUP-22074  |
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| UDC: | 519.17 |
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| eISSN: | 2590-9770 |
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| DOI: | 10.26493/2590-9770.1752.c31 |
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| Publication date in RUP: | 03.11.2025 |
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| Views: | 212 |
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| Downloads: | 2 |
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