Chi_c of line graphs

Question 1      What are the possible values of the circular chromatic number of line graphs ?

Question 2       Does there exist  a 2-edge connected cubic graph G whose line graph L(G)  has circular chromatic number chi_c( L(G) ) > 11/3 ?

The line graph of the Petersen graph has circular chromatic number 11/3.  A negative answer to Question 2 is implies by the Petersen coloring conjecture of Jaeger.

The Petersen coloring conjecture  [Jaeger]      If  G is a 2-edge connected cubic graph, then L(G) admits a homomorphism to L(P), where P is the Petersen graph.

Question 2 has been answered in the negative in [AGGHTZ].  It is proved in [AGGHTZ] that every 2-edge connected graph G of maximum degree at most 3 has chi_c(L(G)) =< 11/3,  with two exceptions H1 and H2. Where H2 is obtained from the complete graph K4 by subdividing one edge, and H1 is obtained as follows: Let K be the graph with two vertices and three parallel edges. Then H1 is obtained from K by subdividing one edge,

To replace Question 2, we ask the following questions:

Question 3 Is it true that for each integer k > 1, there is an epsilon > 0 such that for any line graph L(G), either chi_c(L(G)) >= k  or chi_c(L(G)) =< k - epsilon ?

For k=2, 3, 4, the epsilon exist, and the maximum such epsilon for k=2, 3, 4 are 1, 1/2, 1/3, respectively.  So if the answer to Question 3 is positive, then the next question is whether epsilon can be chosen as 1/(k-1). 

Question 4 If the answer to Question 3 is positive,  is it true that the maximum value of such an epsilon is 1/(k-1) ?

For k=3, C_5 is a graph with chi_c(L(C_5)) = 3- 1/2.  For k =4, the Petersen graph G is a graph with chi_c(L(G)) = 4- 1/3.  As part of Question 4, we

 Question 5 For k >=5, is there a graph G  with chi_c(L(G)) = k- 1/(k-1) ?