6. Graphs

2016 ◽  
pp. 86-108
Author(s):  
Robin Wilson
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Travelling Salesman Problem ◽  
Chemical Bonds ◽  
Hamiltonian Graphs ◽  
Road Map ◽  
Cycle Graph

Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits the Königsberg bridges problem, the knight’s tour problem, the Gas–Water–Electricity problem, the map-colour problem, the minimum connector problem, and the travelling salesman problem and explains how they can all be considered as problems in graph theory. It begins with an explanation of a graph and describes the complete graph, the complete bipartite graph, and the cycle graph, which are all simple graphs. It goes on to describe trees in graph theory, Eulerian and Hamiltonian graphs, and planar graphs.

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

COMPLETELY DISTINGUISHABLE PROJECTIONS OF SPATIAL GRAPHS

2006 ◽  
Vol 15 (01) ◽  
pp. 11-19 ◽  
Author(s):  
RYO NIKKUNI
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Double Point ◽  
Complete Bipartite Graph ◽  
Finite Graph ◽  
Double Points ◽  
Spatial Graphs ◽  
Generic Immersion

A generic immersion of a finite graph into the 2-space with p double points is said to be completely distinguishable if any two of the 2p embeddings of the graph into the 3-space obtained from the immersion by giving over/under information to each double point are not ambient isotopic in the 3-space. We show that only non-trivializable graphs and non-planar graphs have a non-trivial completely distinguishable immersion. We give examples of non-trivial completely distinguishable immersions of several non-trivializable graphs, the complete graph on n vertices and the complete bipartite graph on m + n vertices.

The Coarseness of the Complete Bipartite Graph

1969 ◽  
Vol 21 ◽  
pp. 1086-1096 ◽  
Author(s):  
Lowell W. Beineke ◽  
Richard K. Guy
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
The Other ◽  
Integer Part ◽  
Edge Disjoint ◽  
Image Position

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n, consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.Our results are summarized in the following theorem, where square brackets denote “integer part”.THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1,1

scholarly journals A Note on Pair Sum Graphs

2011 ◽  
Vol 3 (2) ◽  
pp. 321-329 ◽  
Author(s):  
R. Ponraj ◽  
J. X. V. Parthipan ◽  
R. Kala
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Edge Function ◽  
Injective Map ◽  
Cycle Path

Let G be a (p,q) graph. An injective map ƒ: V (G) →{±1, ±2,...,±p} is called a pair sum labeling if the induced edge function, ƒe: E(G)→Z -{0} defined by ƒe (uv)=ƒ(u)+ƒ(v) is one-one and ƒe(E(G)) is either of the form {±k1, ±k2,…, ±kq/2} or {±k1, ±k2,…, ±k(q-1)/2} {k (q+1)/2} according as q is even or odd. Here we prove that every graph is a subgraph of a connected pair sum graph. Also we investigate the pair sum labeling of some graphs which are obtained from cycles. Finally we enumerate all pair sum graphs of order ≤ 5.Keywords: Cycle; Path; Bistar; Complete graph; Complete bipartite graph; Triangular snake.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i2.6290                 J. Sci. Res. 3 (2), 321-329 (2011)

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

2020 ◽  
Vol 30 (1) ◽  
pp. 7-22
Author(s):  
Boris A. Pogorelov ◽  
Marina A. Pudovkina
Keyword(s):  
Vector Space ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Isometry Group ◽  
Complete Bipartite Graph ◽  
Dimensional Vector ◽  
Translation Group ◽  
Dimensional Vector Space ◽  
Hamming Metric

AbstractThe Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

scholarly journals Almost Every Tree With m Edges Decomposes K2m,2m

2013 ◽  
Vol 23 (1) ◽  
pp. 50-65 ◽  
Author(s):  
M. DRMOTA ◽  
A. LLADÓ
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Random Tree ◽  
Equal Size

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size.

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

2009 ◽  
Vol 86 (1) ◽  
pp. 111-122 ◽  
Author(s):  
CAI HENG LI ◽  
JIANGMIN PAN ◽  
LI MA
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Complete Graph ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Prime Power Order ◽  
Normal Cayley Graph ◽  
Normal Cover ◽  
Vertex Transitive

AbstractLet Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

THE TWO-ARC-TRANSITIVE GRAPHS OF SQUARE-FREE ORDER ADMITTING ALTERNATING OR SYMMETRIC GROUPS

2017 ◽  
Vol 104 (1) ◽  
pp. 127-144
Author(s):  
GAI XIA WANG ◽  
ZAI PING LU
Keyword(s):  
Finite Group ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Symmetric Groups ◽  
Transitive Graph ◽  
A Finite Group ◽  
Transitive Graphs ◽  
Free Order

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.

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