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.
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