Self-orthogonal Hamilton path decompositions
Discrete Mathematics - Special volume: Designs and Graphs
On Orthogonal Double Covers of Graphs
Designs, Codes and Cryptography
Minimum matrix representation of closure operations
Discrete Applied Mathematics
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Let $q\equiv 1\textrm{ (mod 4)}$ be a prime power. For a primitive element α of $\mathbb{F}:=\textrm{GF}(q)$ and an element $0\ne t\in\mathbb{F}$ we define a graph G=G(α,t) on the vertex set $\mathbb{F}$ by E(G)=E1∪E2, where $E_1=\left\{\{\alpha^{2i},\alpha^{2i+1}\}\mid i=0,1,\dots, (q-3)/2\right\},$ $E_2=\left\{\{\alpha^{2i+1}+t,\alpha^{2i+2}+t\}\mid i=0,1,\dots, (q-3)/2\right\}.$ It is easy to verify that E1∩E2=∅. Hence, all vertices of G have degree two, except 0 and t which have degree one. In other words, one component of G is a path connecting 0 and t, and all other components are cycles (of even length). Moreover, if $0\ne t,t'\in\mathbb{F}$, then G(α,t) and G(α,t′) are isomorphic [4], justifying the notation G(α).