Homomorphisms of planar (m,n)-colored-mixed graphs to planar targets

An $(m,n)$-colored-mixed graph $G=(V,A_1,A_2,\cdots ,A_m,E_1,E_2,\cdots ,E_n)$ is a graph having m colors of arcs and n colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph G to another $(m,n)$-colored-mixed graph H is a morphism $\phi :V(G)\to V(H)$ such that each edge (resp. arc) of G is mapped to an edge (resp. arc) of H of the same color (and orientation). An $(m,n)$-colored-mixed graph T is said to be $P_g^{(m,n)}$-universal if every graph in $P_g^{(m,n)}$ (the planar $(m,n)$-colored-mixed graphs with girth at least g) admits a homomorphism to T.We show that planar $P_g^{(m,n)}$-universal graphs do not exist for $2m+n\ge 3$ (and any value of g) and find a minimal (in the number vertices) planar $P_g^{(m,n)}$-universal graphs in the other cases.     

Cyclotomic graphs and perfect codes

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[{\zeta }_m]/A$, with connection sets $\{\pm ({\zeta }_m^i+A):0\le i\le m?1\}$ and $\{\pm ({\zeta }_m^i+A):0\le i\le ?(m)?1\}$, respectively, where ${\zeta }_m$ ($m\ge 2$) is an mth primitive root of unity, A a nonzero ideal of $\mathbb{Z}[{\zeta }_m]$, and ? Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by $G_m(A)$ and $G_m^{?}(A)$, respectively. We give a necessary and sufficient condition for $D/A$ to be a perfect t-code in $G_m^{?}(A)$ and a necessary condition for $D/A$ to be such a code in $G_m(A)$, where $t\ge 1$ is an integer and D an ideal of $\mathbb{Z}[{\zeta }_m]$ containing A. In the case when $m=3,4$, $G_m((\alpha ))$ is known as an Eisenstein–Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for $(\beta )/(\alpha )$ to be a perfect t-code in $G_m((\alpha ))$, where $0\ne \alpha ,\beta \in \mathbb{Z}[{\zeta }_m]$ with β dividing α. In the literature such conditions are known to be sufficient when $m=4$ and $m=3$ under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.