## Measurements of edge-uncolorability.(English)Zbl 1041.05031

Summary: 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 $$r_2=\frac{2}{3}|E|-c_2$$. 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 $$\omega$$ 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 $$\omega$$ are arbitrarily far apart, and show that for each $$1 \leqslant c < 2$$ there is a cubic graph such that $$\omega \geqslant cr_3$$. For $$k=2,3$$, let $$\zeta_k$$ 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.

### MSC:

 05C15 Coloring of graphs and hypergraphs
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### References:

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