Abstract
Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k=2,3, let c k be the maximum size of a k-colorable subgraph of a cubic graph G=( V, E). We consider r 3=| E|− c 3 and r 2= 2 3 |E|−c 2 . We show that on one side r 3 and r 2 bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G, the smallest possible number of odd circuits in a 2-factor of G. We construct cyclically 5-edge connected cubic graphs where r 3 and ω are arbitrarily far apart, and show that for each 1⩽ c<2 there is a cubic graph such that ω⩾ cr 3. For k=2,3, let ζ k denote the largest fraction of edges that can be k-colored. We give best possible bounds for these parameters, and relate them to each other.
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.
