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

A \emph{$k$--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 have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs

REDUCTION OF THE BERGE-FULKERSON CONJECTURE TO CYCLICALLY 5-EDGE-CONNECTED SNARKS

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (3-valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non-3-edge-colorable cubic graph. In contrast to Tutte's 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5-edge-connected

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges

Edge-colorings of 4-regular graphs with the minimum number of palettes

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Hor\u148\ue1k, Kalinowski, Meszka and Wo\u17aniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs

A Characterization of Graphs with Small Palette Index

Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of G. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index 1 are r-regular graphs admitting an r-edge-coloring, while regular graphs with palette index 2 do not exist. Here, we characterize all graphs with palette index either 2 or 3 in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index 3

Covering cubic graphs with matchings of large size

Abstract Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called the excessive [m]-index of G in the literature. The case m = n, that is, a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some detail the case m = n − 1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has excessive [n − 1]-index at most 4. Finally, we discuss the relation between excessive [n − 1]-index and some other graph parameters such as oddness and circumference

k\u2013Pyramidal One\u2013Factorizations

We consider one\u2013factorizations of complete graphs which possess an automorphism group fixing k 65 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k>=2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k 12 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k 12 1 involutions, a one\u2013factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed

Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations

Brualdi and Hollingswort conjectured in Brualdi and Hollingsworth (1996) that in any completegraph K 2 n , n 65 3,whichisproperlycoloredwith2 n 12 1colors,theedgesetcanbe partitionedinto n edgedisjointrainbowspanningtrees(whereagraphissaidtoberainbow ifitsedgeshavedistinctcolors).Constantine(2002)strengthenedthisconjectureaskingthe rainbowspanningtreestobepairwiseisomorphic.Healsoshowedanexamplesatisfying hisconjectureforevery2 n 08{ 2 s : s 65 3 } 2a{ 5 \ub7 2 s , s 65 1 } .Caughmann,KrusselandMahoney (2017)recentlyshowedafirstinfinitefamilyofedgecoloringsforwhichtheconjectureof BrualdiandHollingsworthcanbeverified.Inthepresentpaper,weextendthisresulttoall edge-coloringsarisingfromcyclic1-factorizationsof K 2 n constructedbyHartmanandRosa (1985).Finally,weremarkthatourconstructionspermittoextendConstatine\u2019sresultalso toall2 n 08{ 2 s d : s 65 1 , d > 3odd }

Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations

Brualdi and Hollingswort conjectured in Brualdi and Hollingsworth (1996) that in any completegraph K 2 n , n ≥ 3,whichisproperlycoloredwith2 n − 1colors,theedgesetcanbe partitionedinto n edgedisjointrainbowspanningtrees(whereagraphissaidtoberainbow ifitsedgeshavedistinctcolors).Constantine(2002)strengthenedthisconjectureaskingthe rainbowspanningtreestobepairwiseisomorphic.Healsoshowedanexamplesatisfying hisconjectureforevery2 n ∈{ 2 s : s ≥ 3 }∪{ 5 · 2 s , s ≥ 1 } .Caughmann,KrusselandMahoney (2017)recentlyshowedafirstinfinitefamilyofedgecoloringsforwhichtheconjectureof BrualdiandHollingsworthcanbeverified.Inthepresentpaper,weextendthisresulttoall edge-coloringsarisingfromcyclic1-factorizationsof K 2 n constructedbyHartmanandRosa (1985).Finally,weremarkthatourconstructionspermittoextendConstatine’sresultalso toall2 n ∈{ 2 s d : s ≥ 1 , d > 3odd }