Abstract
The color number c(G) of a cubic graph G is the minimum cardinality of a color class of a proper 4-edge-coloring of G. It is well-known that every cubic graph G satisfies c(G) = 0 if G has a Hamiltonian cycle, and c(G) ≤ 2 if G has a Hamiltonian path. In this paper, we extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves.
Highlights
Edge-colorings of cubic graphs were extensively studied in the past few decades
It is well-known that every cubic graph G satisfies c(G) = 0 if G has a Hamiltonian cycle, and c(G) ≤ 2 if G has a Hamiltonian path. We extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves
Note that the bound c(G) ≤ 2 is best possible: for example, the Petersen graph has a Hamiltonian path and the color number is exactly two. With this relation in mind, we are interested in a property related to Hamiltonicity such that all cubic graphs satisfying that property have a small color number
Summary
Edge-colorings of cubic graphs were extensively studied in the past few decades. Several theorems and conjectures on edge-colorings of cubic graphs were formulated. See [2] for more such measures It is well-known that every cubic graph with a Hamiltonian cycle is 3-edge-colorable, that is, c(G) = 0. This fact can be extended to the following, see [9, p. Note that the bound c(G) ≤ 2 is best possible: for example, the Petersen graph has a Hamiltonian path and the color number is exactly two. With this relation in mind, we are interested in a property related to Hamiltonicity such that all cubic graphs satisfying that property have a small color number.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
