Article in volume
Authors:
Danilo Artigas
RIC, Universidade Federal Fluminense, Rio das Ostras, Brazil
email: daniloartigas@id.uff.br
Title:
On the edge-sum distinguishing game
PDFSource:
Discussiones Mathematicae Graph Theory 44(4) (2024) 1449-1469
Received: 2022-11-25 , Revised: 2023-06-07 , Accepted: 2023-06-11 , Available online: 2023-07-08 , https://doi.org/10.7151/dmgt.2502
Abstract:
The Edge-Sum Distinguishing game (ESD game) is a graph labeling game
proposed by Tuza in $2017$. In such a game, the players, traditionally called
Alice and Bob, alternately assign an unused label $f(v) \in \{1,\ldots, s\}$ to
an unlabeled vertex $v$ of a graph $G$, and the induced edge label $\phi(uv)$ of
an edge $uv \in E(G)$ is given by $\phi(uv) = f(u) + f(v)$. Alice's goal is to
end up with an injective vertex labeling of all vertices of $G$ that induces
distinct edge labels, and Bob's goal is to prevent this. Tuza also posed the
following questions about the ESD game: given a simple graph $G$, for which
values of $s$ can Alice win the ESD game? And if Alice wins the ESD game with
the set of labels $\{1,\ldots, s\}$, can she also win with $\{1,\ldots, s+1\}$?
In this work, we partially answer these questions by presenting bounds on the
number of consecutive non-negative integer labels necessary for Alice to win the
ESD game on general and classical families of graphs.
Keywords:
graph labeling, labeling game, maker-breaker game, edge-sum distinguishing game, combinatorial game
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