, At step i, having already built the graph G i+1 , we build G i as follows. For every edge of weight p i of G i+1 , build two new triangles on it, a (p i , q i , r i )-triangle and a (p i , r i , q i )-triangle. We build the signed bipartite graph G from G 1 as follows: an edge xy of weight l is replaced by two paths of lengths l and 2k ? l. If l is negative, we assign a negative sign to an edge of the path of length l, otherwise we assign a negative sign to the path of length 2k ? l. Using Observation 2.7, one can easily verify that mapping G to B would imply a mapping of the weighted graph G 1 to (K V (B) , ad (B,?) ), The starting graph is the weighted triangle of weights p j , q j , r j

, coloring algorithm for YES instances Assuming that a signed bipartite graph (B, ?) of unbalanced-girth 2k is a YES instance of our algorithm, the next step is to give a polynomial-time algorithm which, for an input (G, ?) (a signed bipartite K 4minor-free graph of unbalanced-girth at least 2k) gives a homomorphism to (B, ?). The next algorithm does exactly this using the weighted graph

, Input: ( B, ad (B,?) ), the YES certificate of (B, ?) and a signed bipartite K 4-minor-free graph

.. .. , n do 5: Let x i and y i be the two neighbors of v i in v 1 , v 2 ,. .. v i. 6: In ( B, ad (B,?) ), find a vertex z whose algebraic distances from ?(x i ) and ?(y i ) is either the same as the distances of v i from x i and y i (respectively) or it is their opposites, Compute the signed distance function ad, vol.2

G. , ?. )-to, (. , and ?. , signed) graphs. We would also like to point out that any signed bipartite graph of unbalanced-girth 2k which bounds SBSPG 2k must be of order at least quadratic in k. In an ongoing work, He, Naserasr and Sun have built signed bipartite K 4-minor-free graphs of unbalanced-girth 2k for which any homomorphic image of unbalanced-girth 2k would need ?(k 2 ) vertices. Conjecture 7.2 is believed to be true even for some larger classes of graphs such as the class of K 5minor-free graphs. A notable subclass of K 5-minor-free graphs is the class of partial 3-trees. While we plan to address this in forthcoming works, we point out that a signed bipartite partial 3-tree of unbalanced-girth 2k is built in [10] for which any homomorphic image that is of unbalanced-girth 2k has at least 2 2k?1 vertices. The construction in that work is presented as a planar graph, the planar (H, ?)-coloring problem takes as an input a signed planar graph (G, ?) and outputs YES if there is a homomorphism of

, E(K 4 ))-minor-free signed graphs (or oddK 4-minor-free graphs). In particular, we would like to ask if in our main theorem (Theorem 4.1), the condition of "no K 4-minor" for the underlying graph can be replaced by "no (K 4 , E(K 4 ))-minor" for the signed graph itself. References [1] L. Beaudou, F. Foucaud and R. Naserasr. Homomorphism bounds and edge-colourings of K 4-minorfree graphs, This motivates further studies of the simpler class of (K, vol.4, pp.128-164, 2017.

R. C. Brewster, F. Foucaud, P. Hell, and R. Naserasr, The complexity of signed graph and edgecoloured graph homomorphisms, Discrete Mathematics, vol.340, issue.2, pp.223-235, 2017.

R. J. Duffin, Topology of series-parallel networks, Journal of Mathematical Analysis and Applications, vol.10, pp.303-318, 1965.

M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems, Theoretical Computer Science, vol.1, issue.3, pp.237-267, 1976.

B. Guenin, Packing odd circuit covers: a conjecture, 2005.

R. Naserasr, Homomorphisms and bounds, Canada, 2003.

R. Naserasr, Homomorphisms and edge-coloring of planar graphs, Journal of Combinatorial Theory, Series B, vol.97, issue.3, pp.394-400, 2007.

R. Naserasr, E. Rollová, and . Sopena, Homomorphisms of planar signed graphs to signed projective cubes, Discrete Mathematics & Theoretical Computer Science, vol.15, issue.3, pp.1-12, 2013.

R. Naserasr, E. Rollová, and . Sopena, Homomorphisms of signed graphs, Journal of Graph Theory, vol.79, issue.3, pp.178-212, 2015.

R. Naserasr, S. Sen, and Q. Sun, Walk-powers and homomorphism bounds of planar signed graphs, Graphs and Combinatorics, vol.32, issue.4, pp.1505-1519, 2016.

P. D. Seymour, Colouring series-parallel graphs, Combinatorica, vol.10, issue.4, pp.379-392, 1990.

J. A. Wald and C. J. Colbourn, Steiner trees, partial 2-trees, and minimum IFI networks, Networks, vol.13, pp.159-167, 1983.

T. Zaslavsky, Signed graphs, Discrete Applied Mathematics, vol.4, issue.1, pp.47-74, 1982.