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A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$ . The connected domatic number of $G$ is the maximum number of pairwise disjoint, connected dominating sets in $V (G)$ . We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.

Connected simple graphs and a selection problem

F. Burton Jones (1975)

Czechoslovak Mathematical Journal

Connection on differential modules

Kazimierz Cegiełka (1984)

Banach Center Publications

Constructing and forbidding automorphisms in lifted maps

Dan Archdeacon, Pavol Gvozdjak, Jozef Širáň (1997)

Mathematica Slovaca

Constructing regular maps and graphs from planar quotients

Stanislav Jendroľ, Roman Nedela, Martin Škoviera (1997)

Mathematica Slovaca

Convex 4-valent polytopes with prescribed types of faces

Marián Trenkler (1984)

Commentationes Mathematicae Universitatis Carolinae

Convex Representations of Maps on the Torus and Other Flat Surfaces.

Bojan Mohar (1994)

Discrete & computational geometry

Counting Lamé differential operators

Răzvan Liţcanu (2002)

Rendiconti del Seminario Matematico della Università di Padova

Crossing numbers and cutwidths.

Djidjev, Hristo N., Vrt'o, Imrich (2003)

Journal of Graph Algorithms and Applications

Crossings, colorings, and cliques.

Albertson, Michael O., Cranston, Daniel W., Fox, Jacob (2009)

The Electronic Journal of Combinatorics [electronic only]

Curvature on a graph via its geometric spectrum

Paul Baird (2013)

Actes des rencontres du CIRM

We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Mean curvature is the most natural curvature that arises in this context and corresponds to local liftings of the graph into a suitable Euclidean space. We discuss some examples.

Cycle and path embedding on 5-ary N-cubes

Tsong-Jie Lin, Sun-Yuan Hsieh, Hui-Ling Huang (2009)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study two topological properties of the 5-ary $n$ -cube $Q_{n}^{5}$ . Given two arbitrary distinct nodes $x$ and $y$ in $Q_{n}^{5}$ , we prove that there exists an $x$ - $y$ path of every length ranging from $2 n$ to $5^{n} - 1$ , where $n \geq 2$ . Based on this result, we prove that $Q_{n}^{5}$ is 5-edge-pancyclic by showing that every edge in $Q_{n}^{5}$ lies on a cycle of every length ranging from $5$ to $5^{n}$ .

Cycle and Path Embedding on 5-ary N-cubes

Tsong-Jie Lin, Sun-Yuan Hsieh, Hui-Ling Huang (2008)

RAIRO - Theoretical Informatics and Applications

We study two topological properties of the 5-ary n-cube $Q_{n}^{5}$ . Given two arbitrary distinct nodes x and y in $Q_{n}^{5}$ , we prove that there exists an x-y path of every length ranging from 2n to 5n - 1, where n ≥ 2. Based on this result, we prove that $Q_{n}^{5}$ is 5-edge-pancyclic by showing that every edge in $Q_{n}^{5}$ lies on a cycle of every length ranging from 5 to 5n.

Cycle Double Covers of Infinite Planar Graphs

Mohammad Javaheri (2016)

Discussiones Mathematicae Graph Theory

In this paper, we study the existence of cycle double covers for infinite planar graphs. We show that every infinite locally finite bridgeless k-indivisible graph with a 2-basis admits a cycle double cover.

Cyclic projective planes and Wada dessins.

Streit, Manfred, Wolfart, Jürgen (2001)

Documenta Mathematica

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