Induced Cycles of Fixed Length

Transcription

1 Induced Cycles of Fixed Length Terry McKee Wright State University Dayton, Ohio USA Cycles in Graphs Vanderbilt University 31 May 2012

2 Overview 1. Investigating the fine structure of induced cycles that consist of particular vertices (or edges). 2. Characterizing the graphs that only have induced k-cycles.

3 Given vertices v 1,,v k, define IC v 1,,v k to mean that each v i is adjacent to v i+1 and v k is adjacent to v 1 and no other pairs v i,v j are adjacent. So: IC v 1,,v k holds precisely when v 1,,v k, in the order listed, form an Induced Cycle. Given edges e 1,,e k, define IC e 1,,e k to mean that each e i is adjacent to e i+1 and e k is adjacent to e 1 and no other pairs e i,e j are adjacent. So: IC e 1,,e k holds precisely when e 1,,e k, in the order listed, form an induced cycle except when k = 3: IC e 1,e 2,e 3 holds precisely when e 1,e 2,e 3 form either a triangle or a star.

4 Lemma 1: The following are equivalent for every graph and every k 3: (1.1) Every induced cycle is a k-cycle. (1.2) Every induced hamiltonian subgraph contains vertices v 1,,v k such that IC v 1,,v k. (1.3) Every induced 2-connected subgraph contains vertices v 1,,v k such that IC v 1,,v k. (1.4) Every induced hamiltonian subgraph has girth k.

5 Lemma 3: The following are equivalent for every graph: (3.0v) For every induced hamiltonian subgraph H, there exist v 1,v 2,v 3 V(H) such that IC v 1,v 2,v 3. (3.0e) For every induced hamiltonian subgraph H, there exist e 1,e 2,e 3 E(H) such that IC e 1,e 2,e 3. (3.1v) For every induced hamiltonian subgraph H and every v 1 V(H), there exist v 2,v 3 V(H) such that IC v 1,v 2,v 3. (3.1e) For every induced hamiltonian subgraph H and every e 1 E(H), there exist e 2,e 3 E(H) such that IC e 1,e 2,e 3. (3.2v) For every induced hamiltonian subgraph H and adjacent v 1,v 2 V(H), there exists v 3 V(H) such that IC v 1,v 2,v 3. (3.2e) For every induced hamiltonian subgraph H and adjacent e 1,e 2 E(H), there exists e 3 E(H) such that IC e 1,e 2,e 3. But every two adjacent vertices are in a 4-cycle every two adjacent edges are in a 4-cycle, because of.

6 Lemma 4: If every induced cycle is a k-cycle, then every cycle has length 2 (modulo k 2). Theorem 5: Each of the following is equivalent to every induced cycle of a graph G being a 3-cycle: (5) G is chordal (meaning that every cycle long enough to have a chord, does have a chord). (5v) In every induced hamiltonian subgraph of G, every two adjacent vertices are adjacent to a third vertex. (5e) In every induced hamiltonian subgraph of G, every two adjacent edges are adjacent to a third edge.

7 Define a graph to be a C k -tree recursively as follows: every cycle C k is a C k -tree; every graph obtained by identifying an edge of a C k -tree with an edge of a new copy of C k is a C k -tree. (sometimes called a tree of k-cycles) ((C 3 -trees traditionally called 2-trees)) Theorem 7: If k 5 is odd, then each of the following is equivalent to every induced cycle of a graph G being a k-cycle: (7) Every block of G is a C k -tree. (7v) In every induced hamiltonian subgraph of G, every vertex is contained in a k-cycle. (7e) In every induced hamiltonian subgraph of G, every edge is contained in a k-cycle.

8 Define Θ d k (d 2 and k even) to be the graph that consists of two degree-d vertices connected by d internally-disjoint induced length-k/2 paths. [ so C k = Θ 2 k ] Define a graph to be a Θ k -tree recursively as follows: every graph Θ d k is a Θ k -tree whenever d 2; every graph obtained by identifying an edge of a Θ k -tree with an edge of a new copy of Θ d k (for any d 2) is a Θ k -tree. Two Θ4-trees, each made from one Θ 3 4 graph and one Θ 6 4 graph. (Θ 4 -trees are precisely the 2-connected series-parallel chordal bipartite graphs [2005].)

9 Define Θ d k (d 2 and k even) to be the graph that consists of two degree-d vertices connected by d internally-disjoint induced length-k/2 paths. [ so C k = Θ 2 k ] Define a graph to be a Θ k -tree recursively as follows: every graph Θ d k is a Θ k -tree whenever d 2; every graph obtained by identifying an edge of a Θ k -tree with an edge of a new copy of Θ d k (for any d 2) is a Θ k -tree. Theorem 8: If k 6 is even, then each of the following is equivalent to every induced cycle of a graph G being a k-cycle: (8) Every block of G is a Θ k -tree. (8v) In every induced hamiltonian subgraph of G, every vertex is contained in a k-cycle. (8e) In every induced hamiltonian subgraph of G, every edge is contained in a k-cycle.

10 Summary: Every induced cycle of G is a k-cycle if and only if: k = 3: G is chordal. In every induced hamiltonian subgraph, every two adjacent elements are adjacent to a third. k = 4: G is chordal bipartite. In every induced hamiltonian subgraph, every element is contained in a k-cycle. odd k 5: Every block of G is a C k -tree. In every induced hamiltonian subgraph, every element is contained in a k-cycle. even k 6: Every block of G is a Θ k -tree. In every induced hamiltonian subgraph, every element is contained in a k-cycle. where element means vertex or element means edge