Induced Cycles of Fixed Length Terry McKee Wright State University Dayton, Ohio USA terry.mckee@wright.edu Cycles in Graphs Vanderbilt University 31 May 2012
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.
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.
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.
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.
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.
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.
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].)
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.
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