A reformulation of the four color theorem is to say that $K_4$ is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, $PC(2k)$, (that is the Cayley graph (${\mathbb{Z}}_2^{2k},\{e_1,e_2,\dots ,e_{2k},J\}$, where the $e_i$'s are the standard basis and $J=e_1+e_2+\cdots +e_{2k}$) is a smallest graph of odd-girth $2k+1$ to which every planar graph of odd-girth at least $2k+1$ admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that ${\chi }^{\prime }(G)=\lceil {\chi }_f^{\prime }(G)\rceil$ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.

Both these conjectures are believed to be true for the larger class of $K_5$-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of $K_4$-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth $2k+1$ to admit a homomorphism from any partial t-tree of odd-girth at least $2k+1$. Applying our results to the class of partial 3-trees, which is a rich subclass of $K_5$-minor-free graphs, we prove that $PC(2k)$ is in fact a smallest graph of odd-girth $2k+1$ to which every partial 3-tree of odd-girth at least $2k+1$ admits a homomorphism. We then apply this result to show that every planar $(2k+1)$-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is $2k+1$, is $(2k+1)$-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.

四色定理的重新表述是 ${ķ}_4$是每个平面（无环）图均允许同态的最小图。扩展这个定理，第二作者证明了（使用四色定理）Clebsch图（在16个顶点上众所周知的无三角形图）是最小的图，每个无三角形的平面图都允许同构。作为进一步的概括，他提出了尺寸为2 k的射影立方，$PC（2ķ）$，（即Cayley图（${\mathbb{ž}}_2^{2ķ}，\{{Ë}_{\mathrm{1个}}，{Ë}_2，\dots ，{Ë}_{2ķ}，Ĵ\}$，其中 ${Ë}_{\mathrm{一世}}$是标准依据， $Ĵ={Ë}_{\mathrm{1个}}+{Ë}_2+\cdots +{Ë}_{2ķ}$）是奇数周长的最小图 $2ķ+\mathrm{1个}$ 至少每个奇数周长的平面图 $2ķ+\mathrm{1个}$承认同态。这个猜想与P. Seymour的猜想有关，后者认为平面多重图的分数边色数决定了它的边色数（更准确地说，Seymour猜想是${\chi }^{\prime }（G）=\lceil {\chi }_F^{\prime }（G）\rceil$对于任何平面多重图G）。请注意，西摩猜想对三次（平面）图的限制是泰特对四色定理的重新表述。

这两个推测都被认为对更大的类别是正确的。 ${ķ}_5$-次要无图（包括平面图的类别）。受这些猜想的启发，并扩展了L. Beaudou，F. Foucaud和第二作者的最新工作，该研究研究了类的同态界。${ķ}_4$-次要图，在这项工作中，我们首先给出奇数周长图B的充要条件$2ķ+\mathrm{1个}$至少从奇数周长的任何部分t树接受同态$2ķ+\mathrm{1个}$。将我们的结果应用于部分3树的类，这是3的丰富子类${ķ}_5$-次要图，我们证明 $PC（2ķ）$ 实际上是奇数周长的最小图 $2ķ+\mathrm{1个}$ 奇数周长的每个部分3树至少 $2ķ+\mathrm{1个}$承认同态。然后，我们将这个结果应用于显示每个平面$（2ķ+\mathrm{1个}）$-对偶多图G，其对偶是部分3树，并且其分数边色数为$2ķ+\mathrm{1个}$，是 $（2ķ+\mathrm{1个}）$-边缘可着色。这两个结果都是上述关于四色定理扩展的猜想的一般情况的最著名的支持。