# Equality Perfect Graphs and Digraphs

## Abstract

In the graph colouring game introduced by Bodlaender [7], two players, Alice and Bob, alternately colour uncoloured vertices of a given graph $G$ with one of $k$ colours so that adjacent vertices receive different colours. Alice wins if every vertex is coloured at the end. The game chromatic number of $G$ is the smallest $k$ such that Alice has a winning strategy.

In Bodlaender's original game, Alice begins. We also consider variants of this game where Bob begins or skipping turns is allowed [1] and their generalizations to digraphs [2]. By means of forbidden induced subgraphs (resp.\ forbidden induced subdigraphs), for several pairs $(g_1,g_2)$ of such graph (resp.\ digraph) colouring games $g_1$ and~$g_2$, which define game chromatic numbers $\chi_{g_1}$ and~$\chi_{g_2}$, we characterise the classes of graphs (resp.\ digraphs) such that, for any induced subgraph (resp.\ subdigraph)~$H$, the game chromatic numbers $\chi_{g_1}(H)$ and $\chi_{g_2}(H)$ of~$H$ are equal.

## References

S.D. Andres, Game-perfect graphs. Math. Methods Oper. Res. 69 (2009), 235--250

S.D. Andres, On characterizing game-perfect graphs by forbidden induced subgraphs. Contrib. Discrete Math. 7 (2012), 21--34

S.D. Andres, Game-perfect digraphs. Math. Methods Oper. Res. 76 (2012), 321--341

S.D. Andres and E. Lock, Characterising and recognising game-perfect graphs. Manuscript

T. Bartnicki, J. Grytczuk, H.A. Kierstead, and X. Zhu, The map-coloring game. Amer. Math. Monthly 114 (2007), 793--803

H.L. Bodlaender, On the complexity of some coloring games. Internat. J. Found. Comput. Sci. 2 (1991), 133--147

L. Cai and X. Zhu, Game chromatic index of $k$-degenerate graphs. J. Graph Theory 36 (2001), 144--155

C. Charpentier, The coloring game on planar graphs with large girth, by a result on sparse cactuses. Discrete Math. 340 (2017), 1069--1073

C. Charpentier, B. Effantin, and G. Paris, On the game coloring index of $F^+$-decomposable graphs. Discrete Appl. Math. 236 (2018), 73--83

C. Charpentier and E. Sopena, The incidence game chromatic number of $(a,d)$-decomposable graphs. J. Discrete Algorithms 31 (2015), 14--25

U. Faigle, W. Kern, H. Kierstead, and W.T. Trotter, On the game chromatic number of some classes of graphs. Ars Combin. 35 (1993), 143--150

M.C. Golumbic, Trivially perfect graphs. Discrete Math. 24 (1978), 105--107

D.J. Guan and X. Zhu, Game chromatic number of outerplanar graphs. J. Graph Theory 30 (1999), 67--70

H.A. Kierstead, A simple competitive graph coloring algorithm. J. Combin. Theory Ser. B 78 (2000), 57--68

J.W.H. Liu, The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl. 11 (1990), 134--172

E. Lock, The structure of $g_B$-perfect graphs. Bachelor's thesis, FernUniversitÃ¤t in Hagen, 2016

V. Neumann-Lara, The dichromatic number of a digraph. J. Combin. Theory Ser. B 33 (1982), 265--270

E. Sidorowicz, The game chromatic number and the game colouring number of cactuses, Inform. Process. Lett. 102 (2007), 147--151

E.S. Wolk, The comparability graph of a tree. Proc. Amer. Math. Soc. 13 (1962), 789--795

E.S. Wolk, A note on ``The comparability graph of a tree''. Proc. Amer. Math. Soc. 16 (1965), 17--20

J. Wu and X. Zhu, Lower bounds for the game colouring number of partial $k$-trees and planar graphs. Discrete Math. 308 (2008), 2637--2642

D. Yang and X. Zhu, Game colouring directed graphs. Electron. J. Combin. 17 (2010), R11

X. Zhu, The game coloring number of planar graphs. J. Combin. Theory Ser. B 75 (1999), 245--258

X. Zhu, The game coloring number of pseudo partial $k$-trees. Discrete Math. 215 (2000), 245--262

X. Zhu, Refined activation strategy for the marking game. J. Combin. Theory Ser. B 98 (2008), 1--18

## Downloads

## Published

## Issue

## Section

## License

This copyright statement was adapted from the statement for the University of Calgary Repository and from the statement for the Electronic Journal of Combinatorics (with permission).

The copyright policy for Contributions to Discrete Mathematics (CDM) is changed for all articles appearing in issues of the journal starting from Volume 15 Number 3.

Author(s) retain copyright over submissions published starting from Volume 15 number 3. When the author(s) indicate approval of the finalized version of the article provided by the technical editors of the journal and indicate approval, they grant to Contributions to Discrete Mathematics (CDM) a world-wide, irrevocable, royalty free, non-exclusive license as described below:

The author(s) grant to CDM the right to reproduce, translate (as defined below), and/or distribute the material, including the abstract, in print and electronic format, including but not limited to audio or video.

The author(s) agree that the journal may translate, without changing the content the material, to any medium or format for the purposes of preservation.

The author(s) also agree that the journal may keep more than one copy of the article for the purposes of security, back-up, and preservation.

In granting the journal this license the author(s) warrant that the work is their original work and that they have the right to grant the rights contained in this license.

The authors represent that the work does not, to the best of their knowledge, infringe upon anyoneâ€™s copyright.

If the work contains material for which the author(s) do not hold copyright, the author(s) represent that the unrestricted permission of the copyright holder(s) to grant CDM the rights required by this license has been obtained, and that such third-party owned material is clearly identified and acknowledged within the text or content of the work.

The author(s) agree to ensure, to the extent reasonably possible, that further publication of the Work, with the same or substantially the same content, will acknowledge prior publication in CDM.

The journal highly recommends that the work be published with a Creative Commons license. Unless otherwise arranged at the time the finalized version is approved and the licence granted with CDM, the work will appear with the CC-BY-ND logo. Here is the site to get more detail, and an excerpt from the site about the CC-BY-ND. https://creativecommons.org/licenses/

**Attribution-NoDerivsCC BY-ND**

This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.