Adjacent vertex distinguishing total coloring of the corona product  of graphs

Verma, Shaily; Panda, B. S.

doi:10.7151/dmgt.2445

Cited by 3 publications

(1 citation statement)

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“…If G is the Paley graph P (9) and H = K 1 then tww(G) = 4, tww(H) = 0, and tww(G H) = 5, so tww(G) + 1 is necessary in the maximum. The l-corona product, a repeated product defined as G 1 H = G H and for an integer ℓ ≥ 2, G ℓ H = (G ℓ−1 H) H was recently introduced Furmańczyk and Zuazua (2022). For this repeated product, we derive the following from repeated application of Theorem 3.14.…”

Section: Modular Productmentioning

confidence: 99%

Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

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Section: Modular Productmentioning

confidence: 99%

Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Maker-Breaker resolving game played on corona products of graphs

James,

Klavžar,

Kuziak

et al. 2024

Aequat. Math.

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The Maker-Breaker resolving game is a game played on a graph G by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of G. The goal of Resolver is to select all the vertices in a resolving set of G, while that of Spoiler is to prevent this from happening. The outcome o(G) of the game played is one of $$\mathcal {R}$$ R , $$\mathcal {S}$$ S , and $$\mathcal {N}$$ N , where $$o(G)=\mathcal {R}$$ o ( G ) = R (resp. $$o(G)=\mathcal {S}$$ o ( G ) = S ), if Resolver (resp. Spoiler) has a winning strategy no matter who starts the game, and $$o(G)=\mathcal {N}$$ o ( G ) = N , if the first player has a winning strategy. In this paper, the game is investigated on corona products $$G\odot H$$ G ⊙ H of graphs G and H. It is proved that if $$o(H)\in \{\mathcal {N}, \mathcal {S}\}$$ o ( H ) ∈ { N , S } , then $$o(G\odot H) = \mathcal {S}$$ o ( G ⊙ H ) = S . No such result is possible under the assumption $$o(H) = \mathcal {R}$$ o ( H ) = R . It is proved that $$o(G\odot P_k) = \mathcal {S}$$ o ( G ⊙ P k ) = S if $$k=5$$ k = 5 , otherwise $$o(G\odot P_k) = \mathcal {R}$$ o ( G ⊙ P k ) = R , and that $$o(G\odot C_k) = \mathcal {S}$$ o ( G ⊙ C k ) = S if $$k=3$$ k = 3 , otherwise $$o(G\odot C_k) = \mathcal {R}$$ o ( G ⊙ C k ) = R . Several results are also given on corona products in which the second factor is of diameter at most 2.

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