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### Odd Wheels in Graphs

Baoguang Xu, Guoping Jin, Zhenhong Liu
2002 Journal of combinatorial theory. Series B (Print)
In this paper it is shown that if a graph G of order n with minimum degree greater than 7n/12 is at least 4-chromatic then G contains an odd wheel with at most 5 spokes.  ...  For k \ 1 the odd wheel of 2k+1 spokes, denoted by W 2k+1 , is the graph obtained from a cycle of length 2k+1 by adding a new vertex and joining it to all vertices of the cycle.  ...  ODD WHEELS IN GRAPHS  ...

### On building 4-critical plane and projective plane multiwheels from odd wheels [article]

Dainis Zeps
2012 arXiv   pre-print
We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two.  ...  to odd cycle.  ...  Construction 7 Let us take odd in number (k = 2q + 1) odd wheels and one wheel w q .  ...

### h-perfect plane triangulations [article]

Yohann Benchetrit, Henning Bruhn
2015 arXiv   pre-print
Other minimally t-imperfect graphs are even Möbius ladders, and the circular graph C 2 10 ; see Figure 5 . Only odd wheels will be of relevance in this note.  ...  The loose wheel is a loose odd wheel if C has odd length, and if at least three of the segments are odd as well. Obviously, odd wheels are loose odd wheels.  ...

### On t-perfect triangulations of the projective plane [article]

Elke Fuchs, Laura Gellert
2017 arXiv   pre-print
Evidently, every odd wheel is a loose odd wheel and every graph that contains an odd wheel as a subgraph also contains an odd wheel as an induced subgraph.  ...  Introduction Perfect graphs received considerous attention in graph theory.  ...

### Odd wheels are not odd-distance graphs [article]

Gábor Damásdi
2020 arXiv   pre-print
An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle.  ...  We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane such that the lengths of the edges are odd integers.  ...   on embeddings of wheels. We also thank Dömötör Pálvölgyi and our anonymous reviewers for valuable suggestions and encouragement.  ...

### Even and odd holes in cap-free graphs

Michele Conforti, G�rard Cornu�jols, Ajai Kapoor, Kristina Vu?kovi?
1999 Journal of Graph Theory
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd).  ...  If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable.  ...  In an even wheel (H, x), we can arbitrarily label odd all edges having x as endnode.  ...

### Even and odd holes in cap‐free graphs

Michele Conforti, Gérard Cornuéjols, Ajai Kapoor, Kristina Vušković
1999 Journal of Graph Theory
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd).  ...  If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable.  ...  In an even wheel (H, x), we can arbitrarily label odd all edges having x as endnode.  ...

### Triangle-free graphs that are signable without even holes

Michele Conforti, G�rard Cornu�jols, Ajai Kapoor, Kristina Vu?kovi?
2000 Journal of Graph Theory
We characterize triangle-free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd-signable graphs).  ...  These graphs arise as building blocks of a decomposition theorem (for cap-free odd-signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class.  ...  ACKNOWLEGMENT Ajai Kapoor was supported in part by a grant from Gruppo Nazionale Delle Ricerche-CNR.  ...

### An alternative approach for the anti-magic labelling of a wheel graph and a pendant graph

K. M. P. G. S. C. Kapuhennayake, A. A. I. Perera
2021 Journal of Science
For wheel graph, we removed the middle vertex of the wheel graph and created a path graph using the vertices in the outer cycle of the wheel graph.  ...  Wheel graph is a graph that contains a cycle of length and for which every graph vertex in the cycle is connected to one other graph vertex known as the "hub".  ...  Hence, the wheel graphs is anti-magic. Now consider the vertex summation of each edges in pendant graph, which is shown in Table 2 .  ...

### Francis Guthrie's approach to The Four Color Problem [article]

Asbjørn Brændeland
2018 arXiv   pre-print
The odd wheel is the only type of 4-critical graph in which one vertex always gets a unique color. This supports Frederic Guthrie's approach to the Four Color Problem.  ...  be an odd cycle, and G must be an odd wheel. v Figure 5 For every non-wheel 4-critical graph G and for every vertex v in G there must be another vertex u, independent of v, in G that can be given the  ...  Figure 4 The vertex and the cycle above make out a wheel, and since the cycle is odd, we have an odd wheel, which is the simplest type of 4-critical graph-and also the only type of 4-critical graph in  ...

### Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Rule

Nisha. D, Srividhya. B
2019 Zenodo
In this paper, we consider the wheel graph and also the hamilton graph using if then rules fuzzy numbers.  ...  B "Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Rule" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue  ...  In we consider the wheel graph and also the hamilton graph using fuzzy numbers.  ...

### Reducing quadrangulations of the sphere and the projective plane [article]

Elke Fuchs, Laura Gellert
2017 arXiv   pre-print
The operation is mainly used in the field of t-perfect graphs.  ...  We further show that a non-bipartite quadrangulation of the projective plane can be transformed into an odd wheel by t-contractions and deletions of degree-2 vertices.  ...  A p-wheel W p is a graph consisting of a cycle (w 1 , . . . , w p , w 1 ) and a vertex v adjacent to all vertices of the cycle. A wheel W p is an odd wheel, if p is odd.  ...

### Triangulated neighborhoods in even-hole-free graphs

Murilo V.G. da Silva, Kristina Vušković
2007 Discrete Mathematics
This implies that in an even-hole-free graph, with n nodes and m edges, there are at most n + 2m maximal cliques.  ...  It also yields an O(n 2 m) algorithm that generates all maximal cliques of an even-hole-free graph. In fact these results are obtained for a larger class of graphs that contains even-hole-free graphs.  ...  odd-signable graph in time O(n 2 m).  ...

### Some Results on 2-odd Labeling of Graphs

2020 International journal of recent technology and engineering
In this paper, we investigate 2-odd labeling of some classes of graphs.  ...  A graph is said to be a 2-odd graph if the vertices of can be labelled with integers (necessarily distinct) such that for any two vertices which are adjacent, then the modulus difference of their labels  ...  MAIN RESULTS In this section, we establish 2-odd labeling of some graphs such as wheel graph, fan graph, generalized butterfly graph, the line graph of sunlet graph, and shell graph. Lemma 1.  ...

### On chordal and perfect plane near-triangulations [article]

Sameera M Salam, Daphna Chacko, Nandini J Warrier, K Murali Krishnan, Sudeep K S
2019 arXiv   pre-print
or f induces an odd hole.  ...  The above characterization leads to a linear time algorithm for determining perfectness of this class of graphs.  ...  Definition 2.1 (Wheel). A wheel on n (n ≥ 4) vertices, W n , is the graph obtained by adding a new vertex v to C n−1 and making it adjacent to all vertices in C n−1 .  ...
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