CELLULAR EMBEDDINGS OF LINE GRAPHS AND LIFTS

Labeling Subgraph Embeddings and Cordiality of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Miscellaneous properties of embeddings of line, total and middle graphs

Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characte...

Minors in lifts of graphs

We study here lifts and random lifts of graphs, as defined in [1]. We consider the Hadwiger number η and the Hajós number σ of `lifts of Kn, and analyze their extremal as well as their typical values (that is, for random lifts). When ` = 2, we show that n2 ≤ η ≤ n, and random lifts achieve the lower bound (as n → ∞). For bigger values of `, we show Ω ( n √ logn ) ≤ η ≤ n √ `. We do not know how...

Random Lifts of Graphs III :

Expansion in Lifts of Graphs

The central goal of this thesis is to better understand, and explicitly construct, expanding towers G1,G2, . . ., which are expander families with the additional constraint that Gn+1 is a lift of Gn . A lift G of H is a graph that locally looks like H , but may be globally di erent; lifts have been proposed as a more structured setting for elementary explicit constructions of expanders, and the...