The chromatic number of a graph of girth 5 on a fixed surface

We prove a color extension result implying that, for every fixed surface S, there are only finitely many 4-color-critical graphs of girth 5 on S. The result is best possible in the sense that there are infinitely many 4-color-critical graphs of girth 4 on S. except when S is the sphere. As a consequence, the chromatic number of graphs of girth 5 on S can be found in polynomial time.

[1]  Carsten Thomassen,et al.  A short list color proof of Grötzsch's theorem , 2003, J. Comb. Theory, Ser. B.

[2]  Carsten Thomassen,et al.  Grötzsch's 3-Color Theorem and Its Counterparts for the Torus and the Projective Plane , 1994, J. Comb. Theory, Ser. B.

[3]  Carsten Thomassen,et al.  3-List-Coloring Planar Graphs of Girth 5 , 1995, J. Comb. Theory B.

[4]  Carsten Thomassen,et al.  Color-Critical Graphs on a Fixed Surface , 1997, J. Comb. Theory, Ser. B.

[5]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[6]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[7]  Carsten Thomassen,et al.  Coloring graphs with fixed genus and girth , 1997 .

[8]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[9]  Carsten Thomassen,et al.  Every Planar Graph Is 5-Choosable , 1994, J. Comb. Theory B.

[10]  Barrett Hamilton Walls.  Coloring girth restricted graphs on surfaces , 1999 .

[11]  Bojan Mohar,et al.  Generating locally cyclic triangulations of surfaces , 1992, J. Comb. Theory, Ser. B.

[12]  Bojan Mohar,et al.  Coloring Graphs without Short Non-bounding Cycles , 1994, J. Comb. Theory, Ser. B.

[13]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .