Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

Abstract

A k‐bisection of a bridgeless cubic graph G is a 2‐colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most k. Ban and Linial Conjectured that every bridgeless cubic graph admits a 2‐bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph G with ∣ E ( G ) ∣ ≡ 0 ( mod 2 ) has a 2 ‐edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban‐Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above‐mentioned conjectures. Moreover, we prove Ban‐Linial's Conjecture for cubic‐cycle permutation graphs. As a by‐product of studying 2‐edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

Computational results and new bounds for the circular flow number of snarks

“…Such properties assure that if we construct a flow f starting from f and by adding a flow equal to 1 k along every oriented cycle C i then we obtain a nowhere-zero (4 of S k . Indeed, the former property implies that the edge e has flow value k • 1 k = 1 in f , and the latter one implies that every other edge has flow value in the interval [1, 3 + 1 k ].…”

“…• For k ≥ 3, S k is obtained with a dot product of S k−1 and a copy of the Petersen graph where we select a pair of adjacent vertices of the Petersen graph (by symmetry every pair) and the two independent edges of S k as indicated in Figure 1 (bold edges). It is easy to check that S 2 has order 18 = 8 • 2 + 2 and that it has circular flow number 4 + 1 2 . Now, we assign to every S k an orientation as in Figure 1: the generalization to an arbitrary value of k is clear when we note that any new block with 8 vertices (obtained by dot product with a new copy of the Petersen graph) has the opposite orientation with respect to the last block in S k−1 .…”

“…We would like to stress that if G has circular flow number at least 5, it is not true in general that its flow naturally induces a 2-bisection. For instance, the Petersen graph does not admit a 2-bisection at all, and there exist other bridgeless cubic graphs, admitting a 2-bisection, with the property that every 5-flow does not induce a 2-bisection (see for instance [1,7,21] for a more general discussion about bisections in cubic graphs).…”

“…Therefore ŷ − x ∈[1,4) and if we take E 1 = {x, ŷ} we obtain a graph in H v (G) with circular flow number less than 5.Let us now argue for the case dG (v) ≥ 6. If i∈I x i − j∈J y j ∈ [1, 4),for two suitable subsets I ⊆ {1, .…”

A note on 2-bisections of claw-free cubic graphs

“…Recently, we have given a detailed insight into Ban-Linial's Conjecture (as well as some other related conjectures) in [1], where we also presented theoretical and computational evidence for it. In particular, Ban-Linial's Conjecture is proven for bridgeless cubic graphs in the special case of cycle permutation graphs.…”