Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Some properties of vertex-oblique graphs
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z.
Tuza
Opole University
FIT
Warsaw, 05-06.02.2016
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Introduction
• We consider simple connected graph G = (V , E).
• Neighborhood of a vertex v in graph G is a subgraph of G
induced by the set of vertices adjacent with v .
This subgraph we denote N(v , G).
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Types of irregularity of graphs
• Sedláček’s graphs
Definition
C1 = {G = (V , E) : ∀(u, v ∈ V )(u 6= v ⇒ N(u, G) N(v , G))}.
• Highly irregular graphs
Definition
Graph G is called highly irregular if for every vertex v different
vertices from N(v , G) have different degrees.
• Vertex-oblique
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type.
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type.
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type.
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type.
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
c(4,4)
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type..
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
c(4,4)
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Vertex-oblique graphs
Definition
• The type tG (v ) of a vertex v ∈ V (G) is the ordered
degree-sequence (d1 , . . . , ddG (v ) ).
• A graph G is called vertex-oblique if it contains no two
vertices of the same type.
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
c(4,4)
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Theorem
For every n ­ 6 there exist a vertex-oblique with n vertices.
Construction of vertex-oblique graphs:
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Theorem
For every n ­ 6 there exist a vertex-oblique with n vertices.
Construction of vertex-oblique graphs:
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Theorem
For every n ­ 6 there exist a vertex-oblique with n vertices.
Construction of vertex-oblique graphs:
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Theorem
For every n ­ 6 there exist a vertex-oblique with n vertices.
Construction of vertex-oblique graphs:
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
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Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Theorem
There is no vertex-oblique nontrivial tree.
Corollary
Every vertex-oblique graph contains a cycle.
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Core of graph
A core Gc of a graph G is a subgraph obtained by successive
pruning away all vertices of degree one.
G:
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Core of graph
A core Gc of a graph G is a subgraph obtained by successive
pruning away all vertices of degree one.
Gc :
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Tree with a root
Let G be a graph and v ∈ Gc . G0 is a subgraph of G induced by
the set (V (G)\V (Gc )) ∪ {v }. A tree with root of graph G is a
0
connected component of G (v ) containing v .
G0 :
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E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Lemma 1
Let G be a vertex-oblique graph and v ∈ V (Gc ). If T is a tree of
G with root v and T 6= K1 , then all the following conditions hold.
1
dT (v ) ¬ 2 and for each x ∈ V (T ) \ {v } we have dT (x) ¬ 3.
2
If dT (v ) = 2 (dT (x) = 3 for x ∈ V (T ) \ {v }), then one of the
connected components of T − v (one of the connected
components of T − x, not containing v ) is K1 and the other
one contains a vertex which is a leaf of T and its neighbour
in T is of degree two.
3
T can contain at most one vertex of degree three.
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
Lemma
Let G be a vertex-oblique graph and v ∈ V (Gc ). If T is a tree of
G, then |V (T )| ¬ 9. Moreover, T = K1 or T is one of the 21
trees shown on the picture below.
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
• Problem 1
Are there infinitely many vertex-oblique graphs with
bounded av- erage degree?
• Problem 2
Let a, b be given real numbers and let F(a, b) be the class
of vertex-oblique graphs for which the condition
|E(G)| ¬ a|V (G)| + b holds. Is the class F(a, b) infinite?
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza
Introduction
Types of irregularity of graphs
Vertex-oblique graphs
E. Drgas-Burchardt, K. Kowalska, J. Michael, Z. Tuza Some
properties of vertex-oblique graphs Discrete Mathematics,
339 (2016), 95-102.
Alavi, G. Chartrand, F.R.K. Chung, P. Erdös, R.L. Graham
and O.R. Oellermann Highly irregular graphs, J. Graph
Theory 11(2) (1987), 235–249.
Jírí Sedlǎćek: Lokálni i vlasnoste grafu, Časopis pro
pěstorǎni matematiky 106 (1981), 290–298.
J.Schreyer: Almost every graph is vertex-oblique,, Descrete
Math. 307 (2007) 983–989.
Thank You for attention!
E.Drgas-Burchardt, Kamila Kowalska-Żak, J. Michael, Z. Tuza