1,720,977 research outputs found
An equivalent formulation of the Fan-Raspaud Conjecture and related problems
Get PDFIn 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
Normal 6-edge-colorings of some bridgeless cubic graphs
No full textIn an edge-coloring (proper) of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly three or exactly five distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal k-edge-coloring of a cubic graph is an edge-coloring with k colors such that each edge of the graph is normal. We denote by chi(N)'(G) the smallest k, for which G admits a normal k-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving chi(N)'(G) <= 5 for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal 7-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal 6-edge-coloring. Finally, we show that any bridgeless cubic graph G admits a 6-edge-coloring such that at least 7/9 . vertical bar E vertical bar edges of G are normal. (C) 2019 Elsevier B.V. All rights reserved
REDUCTION OF THE BERGE-FULKERSON CONJECTURE TO CYCLICALLY 5-EDGE-CONNECTED SNARKS
Get PDFThe 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
Normal 5-edge-colorings of a family of Loupekhine snarks
Get PDFIn a proper edge-coloring of a cubic graph an edge uv is called poor or rich, if the set of colors of the edges incident to u and v contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge-coloring with five colors implies the existence of a Berge-Fulkerson Covering, our main theorem can be viewed as a generalization of their result
Edge-colorings of 4-regular graphs with the minimum number of palettes
Get PDFA 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 Horňák, Kalinowski, Meszka and Woźniak, 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
Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations
No full textBrualdi and Hollingsworth conjectured in Brualdi and Hollingsworth (1996) that in any complete graph K2n, n≥3, which is properly colored with 2n−1 colors, the edge set can be partitioned into n edge disjoint rainbow spanning trees (where a graph is said to be rainbow if its edges have distinct colors). Constantine (2002) strengthened this conjecture asking the rainbow spanning trees to be pairwise isomorphic. He also showed an example satisfying his conjecture for every 2n∈{2s:s≥3}∪{5⋅2s,s≥1}. Caughmann, Krussel and Mahoney (2017) recently showed a first infinite family of edge colorings for which the conjecture of Brualdi and Hollingsworth can be verified. In the present paper, we extend this result to all edge-colorings arising from cyclic 1-factorizations of K2n constructed by Hartman and Rosa (1985). Finally, we remark that our constructions permit to extend Constatine's result also to all 2n∈{2sd:s≥1,d>3odd}
A Characterization of Graphs with Small Palette Index
Get PDFGiven 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
Upper bounds for the automorphic chromatic index of a graph
No full textThe automorphic H-chromatic index of a graph G is the
minimum integer m for which G has a proper edge-coloring
with m colors preserved by a given subgroup H of the full
automorphism group of G. We determine upper bounds for this index
in terms of the chromatic index of G for some abelian 2-groups H
Extending perfect matchings to Hamiltonian cycles in line graphs
Get PDFA graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph G in order to guarantee that its line graph L(G) has the PMH-property. In particular, we prove that this happens when G is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper
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