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 / 17
2 Introduction Definition: An eulerian digraph G is a digraph graph that contains a closed walk that visits each edge exactly once. Theorem. A digraph G is eulerian if and only if deg ( ) = deg + ( ) for all vertices and G is strongly (weakly) connected James Carraher (UNL) Compatible circuits in eulerian digraphs 2 / 17
3 Introduction Definition: An eulerian digraph G is a digraph graph that contains a closed walk that visits each edge exactly once. Theorem. A digraph G is eulerian if and only if deg ( ) = deg + ( ) for all vertices and G is strongly (weakly) connected James Carraher (UNL) Compatible circuits in eulerian digraphs 2 / 17
4 Introduction Eulerian digraphs can be applied to routing problems such as garbage collecting, mail carriers, etc. Eulerian undirected graphs can be applied to reconstructing DNA from its segments. James Carraher (UNL) Compatible circuits in eulerian digraphs 3 / 17
5 Compatible Circuits 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). 1 Good Bad James Carraher (UNL) Compatible circuits in eulerian digraphs 4 / 17
6 Compatible Circuits 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). 1 Good Bad James Carraher (UNL) Compatible circuits in eulerian digraphs 4 / 17
7 Compatible Circuits Big Question: When does an colored eulerian digraph have a compatible circuit? Not all graphs have compatible circuits. James Carraher (UNL) Compatible circuits in eulerian digraphs 5 / 17
8 Compatible Circuits Let γ( ) be the size of the largest color class incident to. v James Carraher (UNL) Compatible circuits in eulerian digraphs 6 / 17
9 Compatible Circuits 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 6 / 17
10 Compatible Circuits 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 7 / 17
11 Compatible Circuits 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 7 / 17
12 Compatible Circuits Splitting a vertex where γ( ) = deg + ( ). G G v G' G' v 1 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 8 / 17
13 Compatible Circuits Splitting a vertex where γ( ) = deg + ( ). G G v G' G' v 1 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 8 / 17
14 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 9 / 17
15 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 9 / 17
16 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 10 / 17
17 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 10 / 17
18 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 10 / 17
19 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 11 / 17
20 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 12 / 17
21 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 12 / 17
22 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 13 / 17
23 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 13 / 17
24 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 14 / 17
25 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. G G S A B C A C H G B D D James Carraher (UNL) Compatible circuits in eulerian digraphs 14 / 17
26 Non-fixable vertices Prop. A multigraph H has a rainbow spanning tree iff (# 2-trails in a cut S) + 1 # components in H S. Comment: There is a polynomial time algorithm to determine if a multigraph H contains a rainbow spanning tree. James Carraher (UNL) Compatible circuits in eulerian digraphs 15 / 17
27 Non-fixable vertices Graphs with nonfixable outdegree 3 vertices. James Carraher (UNL) Compatible circuits in eulerian digraphs 16 / 17
28 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 17 / 17
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