For a set S of connected graphs, a spanning subgraph F of a graph is called an Sfactor if every component of F is isomorphic to a member of S. It was recently shown that every 2-connected cubic graph has a {Cn|n ≥ 4}-factor and a {Pn|n ≥ 6}factor, where Cn and Pn denote the cycle and the path of order n, respectively (Kawarabayashi et al., J. Graph Theory, Vol. 39 (2002) 188–193). In this paper...

Abstract: When restricted to the rank 1 and rank 2 faces, the Hasse diagram of a regular abstract 4-polytope provides a bipartite graph with a high degree of symmetry. Focusing on the case of the self-dual polytopes of type 3,q,3, I will show that the graphs obtained are 3-arc transitive cubic graphs. Also, given any 3-arc transitive cubic graph, I will discuss when it is possible to consider t...

We focus on a specific class of planar cubic bridgeless graphs, namely the klee-graphs. K4 is the smallest klee-graph and having a klee-graph G we create another klee-graph by replacing any vertex of G by a triangle, i.e., applying Y∆ operation. In this paper, we prove that every klee-graph with n ≥ 8 vertices has at least 3 · 2 perfect matchings, improving the 2 bound inherited from the genera...

Recently, the first author and his coauthor proved a k-order homogeneous linear recursion for the genus polynomials of any H-linear family of graphs (called path-like graph families by Mohar). Cubic outerplanar graphs are tree-like graph families. In this paper, we derive a recursive formula for the total embedding distribution of any cubic outerplanar graph. We also obtain explicit formulas fo...

A snark is a cyclically-4-edge-connected cubic graph with chromatic index 4. In 1880, Tait proved that the Four-Color Conjecture is equivalent to the statement that every planar bridgeless cubic graph has chromatic index 3. The search for counter-examples to the FourColor Conjecture motivated the definition of the snarks. A k-total-coloring of G is an assignment of k colors to the edges and ver...

After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4/3 conjectu...

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3-edge-coloring such t...

Two edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar ...