Definition:
A graphG= (V,E)is called Eulerian, iff there exists a cycle which visits each edge precisely once.
Theorem
A non-directed graphG= (V,E)is Eulerian G is connected and
each node ofG has even degree.
Theorem
A directed graphG= (V,E)is Eulerian G is strong connected and
each node as as many incoming edges as outgoing ones.
Problem: Compute Eulerian cycle on Eulerian graphs.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:14 Introduction 2/4 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Eulerian cycle
Definition:
A graphG= (V,E)is called Eulerian, iff there exists a cycle which visits each edge precisely once.
Theorem
A non-directed graphG= (V,E)is Eulerian G is connected and
each node ofG has even degree.
Theorem
A directed graphG= (V,E)is Eulerian G is strong connected and
each node as as many incoming edges as outgoing ones.
Problem: Compute Eulerian cycle on Eulerian graphs.
Eulerian cycle
Definition:
A graphG= (V,E)is called Eulerian, iff there exists a cycle which visits each edge precisely once.
Theorem
A non-directed graphG= (V,E)is Eulerian G is connected and
each node ofG has even degree.
Theorem
A directed graphG= (V,E)is Eulerian G is strong connected and
each node as as many incoming edges as outgoing ones.
Problem: Compute Eulerian cycle on Eulerian graphs.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:14 Introduction 4/4 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Eulerian cycle
Definition:
A graphG= (V,E)is called Eulerian, iff there exists a cycle which visits each edge precisely once.
Theorem
A non-directed graphG= (V,E)is Eulerian G is connected and
each node ofG has even degree.
Theorem
A directed graphG= (V,E)is Eulerian G is strong connected and
each node as as many incoming edges as outgoing ones.
Problem: Compute Eulerian cycle on Eulerian graphs.
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:15 Introduction 2/10 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:15 Introduction 4/10 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:15 Introduction 6/10 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:15 Introduction 8/10 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.
Colorings I Eulerian cycle Matchings Colorings II MIS Coloring III S.
3:15 Introduction 10/10 Walter Unger 30.1.2017 11:52 WS2016/17 Z
Idea
Non Parallel:
Start with a free edge and follow free/unused edges till a cycle is closed.
Repeat till all edges are is some cycle.
Join pairs of cycles into a single one.
Repeat till just one cycle remains.
IfG is non-directed, then make a directed version ofG. Compute a cover of cycles.
Compute an additional cycle which meets each cycle precisely once.
Uses these to compute a cycle forG
Delete some edges to get a Eulerian cycle forG.