Measures of edge-uncolorability

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $\Delta$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{\Delta(G)}{2} \rfloor$, and that this bound is best possible.Comment: 9 page

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

The Color Number of Cubic Graphs Having a Spanning Tree with a Bounded Number of Leaves

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

Minimal edge colorings of class 2 graphs and double graphs

A proper edge coloring of a class 2 graph G is minimal if it contains a color class of cardinality equal to the resistance r(G) of G, which is the minimum number of edges that have to be removed from G to obtain a graph which is Δ(G)-edge colorable, where Δ(G) is the maximum degree of G. In this paper using some properties of minimal edge colorings of a class 2 graph and the notion of reflective edge colorings of the direct product of two graphs, we are able to prove that the double graph of a class 2 graph is of class 1. This result, recently conjectured, is moreover extended to some generalized double graphs

The smallest nontrivial snarks of oddness 4

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs admitting no proper 3-edge-colouring). In [Ars Math. Contemp. 16 (2019), 277-298] we have proved that the smallest number of vertices of a snark with cyclic connectivity 4 and oddness 4 is 44. We now show that there are exactly 31 such snarks, all of them having girth 5. These snarks are built up from subgraphs of the Petersen graph and a small number of additional vertices. Depending on their structure they fall into six classes, each class giving rise to an infinite family of snarks with oddness at least 4 with increasing order. We explain the reasons why these snarks have oddness 4 and prove that the 31 snarks form the complete set of snarks with cyclic connectivity 4 and oddness 4 on 44 vertices. The proof is a combination of a purely theoretical approach with extensive computations performed by a computer.Comment: 38 pages; submitted for publicatio