Circular Chromatic Number of Signed Planar Graphs Without Cycles of Length 4 to 9

Given a signed graph ( G , σ ) and a positive real number r , if there exists a vertex mapping c : V ( G ) → [ 0 , r ) satisfying that for every positive edge w x , 1 ≤ | c ( w ) − c ( x ) | ≤ r − 1 and for every negative edge w x , | c ( w ) − c ( x ) | ≤ r 2 − 1 or | c ( w ) − c ( x ) | ≥ r 2 + 1 , then ( G , σ ) admits a circular r -coloring. We use χ c ( G , σ ) to represent the circular chromatic number of ( G , σ ) , which is the minimum r , such that a circular r -coloring of ( G , σ ) exists. This paper proves that χ c ( G , σ ) < 4 , where ( G , σ ) is a simple signed planar graph containing no cycles of length 4 to 9. Moreover, we establish an upper bound for the chromatic number of such a graph to be 4 − 2 ⌊ v ( G ) + 1 2 ⌋ .