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### Embedding graphs having Ore-degree at most five [article]

Béla Csaba, Judit Nagy-György
2018 arXiv   pre-print
Let H and G be graphs on n vertices, where n is sufficiently large. We prove that if H has Ore-degree at most 5 and G has minimum degree at least 2n/3 then H⊂ G.  ...  It is easy to see that at most two vertices can have degree at most 1 in the saturated part of H, so |V (H 1 ) ∩ (D 0 ∪ D 1 )| ≤ 2.  ...  The minimum degree of G r may also decrease by at most 12dℓ ′ .  ...

### Embedding Graphs Having Ore-Degree at Most Five

Béla Csaba, Judit Nagy-György
2019 SIAM Journal on Discrete Mathematics
Let H and G be graphs on n vertices, where n is sufficiently large. We prove that if H has Ore-degree at most 5 and G has minimum degree at least 2n/3 then H is a subgraph of G.  ...  It is easy to see that at most two vertices can have degree at most 1 in the saturated part of H, therefore, we have that |V (H 1 ) ∩ (D 0 ∪ D 1 )| ≤ 2.  ...  Then S ′ contains at most |S| elements that are adjacent to W in Λ 2 and have degree 1.  ...

### Embedding graphs in their complements

R. J. Faudree, C. C. Rousseau, Richard H. Schelp, Seymour Schuster
1981 Czechoslovak Mathematical Journal
Then the nonexistence of cycles of length 3 or 4 imphes that there are at most five edges joining vertices of H to vertices of L.  ...  Thus, since all the vertices of degree 1 in G are adjacent to v, the star L has at most five end vertices. Each of the possibilities force G to have more than (6/5) n -2 edges, a contradiction.  ...

### Coloring of Triangle-Free Graphs on the Double Torus

Daniel Král', Matěj Stehlík
2008 SIAM Journal on Discrete Mathematics
We show that every triangle-free graph on the double torus is 4-colorable. This settles a problem raised by Gimbel and Thomassen [Trans. Amer. Math. Soc. 349 (1997), 4555-4564].  ...  The number of edges of an n-vertex triangle-free graph that can be embedded on the double torus is at most 2n + 4.  ...  Moreover, the degree of each vertex of H 4 (G) in G is five. Since H 4 (G) has eight vertices and seven edges, H 4 (G) contains at least four vertices v of degree at most two.  ...

### Testing Euclidean minimum spanning trees in the plane

Artur Czumaj, Christian Sohler
2008 ACM Transactions on Algorithms
Chazelle et al. [3] presented an algorithm that, given a connected graph in adjacency list representation with average degree D, edge weights in the range [1 . . .  ...  approximation algorithms have been presented for the problem of estimating the weight of the MST.  ...  Lemma 3.15 Let G be a geometric graph with maximum degree at most five.  ...

### LINKING IN STRAIGHT-EDGE EMBEDDINGS OF K7

LEWIS D. LUDWIG, PAMELA ARBISI
2010 Journal of knot theory and its ramifications
In 2004 it was shown that all straightedge embeddings of K 6 have either one or three linked triangle pairs.  ...  In 1983 Conway and Gordon and Sachs proved that every embedding of the complete graph on six vertices, K 6 , is intrinsically linked.  ...  We also thank the referee for his or her invaluable suggestions.  ...

### Node Centralities and Classification Performance for Characterizing Node Embedding Algorithms [article]

Kento Nozawa, Masanari Kimura, Atsunori Kanemura
2018 arXiv   pre-print
or five graph centralities, and six datasets.  ...  We examine the performance of node embedding algorithms with respect to graph centrality measures that characterize diverse graphs, through systematic experiments with four node embedding algorithms, four  ...  The embedding dimensionality d was either 128 or 256. Whether to normalize or not to normalize embeddings before passing them to a classifier.  ...

### A linear algorithm for five-coloring a planar graph [chapter]

N. Chiba, T. Nishizeki, N. Saito
1981 Lecture Notes in Computer Science
A simple linear algorithm is presented for coloring planar graphs with at most five colors.  ...  The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.  ...  Since each reduction produces at most two vertices in S, there must occur at least IsI/2 graph reductions around vertices of degree 5 or 6 in the stage.  ...

### A linear 5-coloring algorithm of planar graphs

Norishige Chiba, Takao Nishizeki, Nobuji Saito
1981 Journal of Algorithms
A simple linear algorithm is presented for coloring planar graphs with at most five colors.  ...  The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors. 317  ...  planar graph G with at most five colors.  ...

### Forbidden minors characterization of partial 3-trees

Stefan Arnborg, Andrzej Proskurowski, Derek G. Corneil
1990 Discrete Mathematics
A k-tree is formed from a k-complete graph by recursively adding a vertex adjacent to all vertices in an existing k-complete subgraph.  ...  The many applications of partial k-trees (subgraphs of k-trees) have motivated their study from both the algorithmic and theoretical points of view.  ...  By Lemma 2.4, we only have to consider partial 3-trees without embeddings containing two separators with at most one vertex in common.  ...

### TopoDetect: Framework for topological features detection in graph embeddings

Maroun Haddad, Mohamed Bouguessa
2021 Software Impacts
in the embeddings of graph representation models.  ...  TopoDetect is a Python package that allows the user to investigate if important topological features, such as the Degree of the nodes, their Triangle Count, or their Local Clustering Score, are preserved  ...  Each Degree range (or class) was colored in a unique color.  ...

### Facial parity edge colouring of plane pseudographs

Július Czap, Stanislav Jendroľ, František Kardoš, Roman Soták
2012 Discrete Mathematics
For G being 4-edge-connected plane graph we have χ ′ p (G) ≤ 9. On the other hand we prove that some bridgeless plane graphs require at least 10 colours for such a colouring.  ...  c, either no edge or an odd number of edges incident with f is coloured with c.  ...  having odd degree.  ...

### Obstructions for two-vertex alternating embeddings of graphs in surfaces [article]

Bojan Mohar, Petr Škoda
2011 arXiv   pre-print
A class of graphs that lies strictly between the classes of graphs of genus (at most) k-1 and k is studied.  ...  For a fixed orientable surface S_k of genus k, let A_xy^k be the minor-closed class of graphs with terminals x and y that either embed into S_k-1 or admit an embedding Π into S_k such that there is a Π-face  ...  (S1) Vertices of degree at most 2 in H are labelled. Leaves in H have both labels.  ...

### Using Cellular Graph Embeddings in Solving All Pairs Shortest Paths Problems

G.N. Frederickson
1995 Journal of Algorithms
The edges have real-valued costs, but the graph contains no negative cycles. The algorithm runs in O('Y(G)n + ('Y(G) )'log 'Y(G)) time, where 'Y( G) is a topological emhedding measure that we define.  ...  An algorithm is presented for generating a succinct encoding of all pairs shortest path information in an n-vertex directed graph G with O(n) edges.  ...  This follows since a face in Jill bounded by at least five edges would have a vertex that is not an attachment or replacement attachment vertex in H" that is of degree 2 in Ga.  ...

### Using cellular graph embeddings in solving all pairs shortest paths problems

G.N. Frederickson
1989 30th Annual Symposium on Foundations of Computer Science
The edges have real-valued costs, but the graph contains no negative cycles. The algorithm runs in O('Y(G)n + ('Y(G) )'log 'Y(G)) time, where 'Y( G) is a topological emhedding measure that we define.  ...  An algorithm is presented for generating a succinct encoding of all pairs shortest path information in an n-vertex directed graph G with O(n) edges.  ...  This follows since a face in Jill bounded by at least five edges would have a vertex that is not an attachment or replacement attachment vertex in H" that is of degree 2 in Ga.  ...
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