Incidence coloring game and arboricity of graphs
Cl\'ement Charpentier (LaBRI), Eric Sopena (LaBRI)

TL;DR
This paper investigates the incidence coloring game on graphs, establishing an improved upper bound for the incidence game chromatic number in terms of maximum degree and arboricity, which tightens previous bounds.
Contribution
The paper introduces a new upper bound for the incidence game chromatic number based on arboricity, refining earlier results and demonstrating the bound's tightness with existing examples.
Findings
New upper bound: $i_g(G) \,\le\, \lfloor\frac{3\Delta(G) - a(G)}{2}\rfloor + 8a(G) - 2$
Bound is tight up to a constant factor, matching known lower bounds
Improves previous bounds for graphs with given arboricity
Abstract
An incidence of a graph is a pair where is a vertex of and an edge incident to . Two incidences and are adjacent whenever , or , or or . The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980-1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alternately color the incidences of a graph, using a given number of colors, in such a way that adjacent incidences get distinct colors. If the whole graph is colored then Alice wins the game otherwise Bob wins the game. The incidence game chromatic number of a graph is the minimum number of colors for which Alice has a winning strategy when playing the incidence coloring game on . Andres proved that %i_g(G) \le 2\Delta(G) + 4k - 2$…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Scheduling and Timetabling Solutions
