Compatible circuits in eulerian digraphs
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1 Compatible circuits in eulerian digraphs James Carraher University of Nebraska Lincoln Joint Work with Stephen Hartke March 2012 James Carraher (UNL) Compatible circuits in eulerian digraphs 1 / 12
2 Introduction Definition: A colored eulerian digraph G is an eulerian digraph with a given edge coloring (not necessarily proper). Definition: A compatible circuit is an eulerian circuit of G such that no two consecutive edges in the tour have the same color (i.e. no monochromatic transitions) James Carraher (UNL) Compatible circuits in eulerian digraphs 2 / 12
3 Introduction Definition: A colored eulerian digraph G is an eulerian digraph with a given edge coloring (not necessarily proper). Definition: A compatible circuit is an eulerian circuit of G such that no two consecutive edges in the tour have the same color (i.e. no monochromatic transitions) James Carraher (UNL) Compatible circuits in eulerian digraphs 2 / 12
4 Introduction Let γ( ) be the size of the largest color class incident to. v James Carraher (UNL) Compatible circuits in eulerian digraphs 3 / 12
5 Introduction Let γ( ) be the size of the largest color class incident to. If G is a colored eulerian digraph and there exists a vertex where γ( ) > deg + ( ), then G does not have a compatible circuit. Example v James Carraher (UNL) Compatible circuits in eulerian digraphs 3 / 12
6 Introduction Theorem [Kotzig, 1968] If G is a colored eulerian undirected graph and γ( ) deg( )/2 then G has a compatible circuit. James Carraher (UNL) Compatible circuits in eulerian digraphs 4 / 12
7 Introduction Theorem [Kotzig, 1968] If G is a colored eulerian undirected graph and γ( ) deg( )/2 then G has a compatible circuit. A colored eulerian digraph with γ( ) deg + ( ) does not necessarily have a compatible circuit. James Carraher (UNL) Compatible circuits in eulerian digraphs 4 / 12
8 Introduction Splitting a vertex where γ( ) = deg + ( ) G G' v 1 v v 2 The graph G has a compatible circuit if and only if the graph G has a compatible circuit. James Carraher (UNL) Compatible circuits in eulerian digraphs 5 / 12
9 Fixable vertices Let T be an eulerian circuit of G and a vertex of G. We define the excursion graph L T ( ) to be the following graph. G v L (v) T v James Carraher (UNL) Compatible circuits in eulerian digraphs 6 / 12
10 Fixable vertices Let T be an eulerian circuit of G and a vertex of G. We define the excursion graph L T ( ) to be the following graph. G v L (v) T v Definition: A vertex is fixable if L M ( ) has a compatible circuit for any matching M between E + ( ) and E ( ). James Carraher (UNL) Compatible circuits in eulerian digraphs 6 / 12
11 Fixable vertices Proposition. Let G be a colored eulerian digraph. If every vertex of G is a fixable vertex then G has a compatible circuit. Idea: Iteratively fix fixable vertices. James Carraher (UNL) Compatible circuits in eulerian digraphs 7 / 12
12 Fixable vertices Proposition. Let G be a colored eulerian digraph. If every vertex of G is a fixable vertex then G has a compatible circuit. Idea: Iteratively fix fixable vertices. Proposition. A vertex with γ( ) < deg + ( ) is fixable if and only if it does not have the form below. James Carraher (UNL) Compatible circuits in eulerian digraphs 7 / 12
13 Fixable vertices Proposition. Let G be a colored eulerian digraph. If every vertex of G is a fixable vertex then G has a compatible circuit. Idea: Iteratively fix fixable vertices. Proposition. A vertex with γ( ) < deg + ( ) is fixable if and only if it does not have the form below. Example The excursion graph L M ( ) does not have a compatible circuit. James Carraher (UNL) Compatible circuits in eulerian digraphs 7 / 12
14 Non-fixable vertices Let S be the set of vertices that are not fixable. Let S 3 be the subset of S with vertices of outdegree three. I.e. We will consider colored eulerian digraphs with no nonfixable vertices of outdegree three. James Carraher (UNL) Compatible circuits in eulerian digraphs 8 / 12
15 Non-fixable vertices Some Auxiliary Graphs The graph G, G S, and component graph H G. G G S A B C A C H G B D D James Carraher (UNL) Compatible circuits in eulerian digraphs 9 / 12
16 Non-fixable vertices Some Auxiliary Graphs The graph G, G S, and component graph H G. G G S A B C A C H G B D D James Carraher (UNL) Compatible circuits in eulerian digraphs 9 / 12
17 Non-fixable vertices Problem: Let H be a multigraph whose edge set is the disjoint union of 2-trails. When does there exist a subset E of the edges such that 1 E contains at most one edge from each 2-trail, and 2 the spanning subgraph with edge set E is connected? James Carraher (UNL) Compatible circuits in eulerian digraphs 10 / 12
18 Non-fixable vertices Problem: Let H be a multigraph whose edge set is the disjoint union of 2-trails. When does there exist a subset E of the edges such that 1 E contains at most one edge from each 2-trail, and 2 the spanning subgraph with edge set E is connected? A multigraph H with the property above will be said to contain a rainbow spanning tree. James Carraher (UNL) Compatible circuits in eulerian digraphs 10 / 12
19 Non-fixable vertices Theorem. Let G be a colored eulerian digraph with no nonfixable vertices of outdegree three. Then G has a compatible circuit if and only if the component graph H G contains a rainbow spanning tree. James Carraher (UNL) Compatible circuits in eulerian digraphs 11 / 12
20 Non-fixable vertices Theorem. Let G be a colored eulerian digraph with no nonfixable vertices of outdegree three. Then G has a compatible circuit if and only if the component graph H G contains a rainbow spanning tree. Comment: There is a polynomial time algorithm to determine if a multigraph H contains a rainbow spanning tree. Therefore if G has no nonfixable vertices of outdegree three then there is a polynomial time algorithm to determine if G has a compatible circuit. James Carraher (UNL) Compatible circuits in eulerian digraphs 11 / 12
21 Non-fixable vertices Theorem. Let G be a colored eulerian digraph with no nonfixable vertices of outdegree three. Then G has a compatible circuit if and only if the component graph H G contains a rainbow spanning tree. Comment: There is a polynomial time algorithm to determine if a multigraph H contains a rainbow spanning tree. Therefore if G has no nonfixable vertices of outdegree three then there is a polynomial time algorithm to determine if G has a compatible circuit. Open Question: Does there exist a polynomial time algorithm to determine if G has a compatible circuit if G contains nonfixable vertices of outdegree three? James Carraher (UNL) Compatible circuits in eulerian digraphs 11 / 12
22 Non-fixable vertices Compatible circuits in eulerian digraphs James Carraher University of Nebraska Lincoln Joint Work with Stephen Hartke March 2012 James Carraher (UNL) Compatible circuits in eulerian digraphs 12 / 12
Compatible circuits in eulerian digraphs
Compatible circuits in eulerian digraphs James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Stephen Hartke March 2012 James Carraher (UNL) Compatible circuits in eulerian
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