Covering cubic graphs with matchings of large size

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 excessive [m]-index of G in 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 details 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 as oddness and circumference.Comment: 11 pages, 5 figure

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

On the existence spectrum for sharply transitive G-designs, G a [k]-matching

In this paper we consider decompositions of the complete graph Kv into matchings of uniform cardinality k. They can only exist when k is an admissible value, that is a divisor of v(v−1)/2 with 1≤k≤v/2. The decompositions are required to admit an automorphism group Γ acting sharply transitively on the set of vertices. Here Γ is assumed to be either non-cyclic abelian or dihedral and we obtain necessary conditions for the existence of the decomposition when k is an admissible value with 1<k<v/2. Differently from the case where Γ is a cyclic group, these conditions do exclude existence in specific cases. On the other hand we produce several constructions for a wide range of admissible values, in particular for every admissible value of k when v is odd and Γ is an arbitrary group of odd order possessing a subgroup of order gcd(k,v)

Even cycles and even 2-factors in the line graph of a simple graph

Let G be a connected graph with an even number of edges. We show that if the subgraph of G induced by the vertices of odd degree has a perfect matching, then the line graph of G has a 2-factor whose connected components are cycles of even length (an even 2-factor). For a cubic graphG, we also give a necessary and sufficient condition so that the corresponding line graph L(G) has an even cycle decomposition of index 3, i.e., the edge-set of L(G) can be partitioned into three 2-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index m in 2d-regular graphs is also addressed

A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I-graphs follows from the fact that one can choose a 1-factor in any I-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I-graphs as a subfamily

Metabolomic analysis and bioactivities of Arbutus unedo leaves harvested across the seasons in different natural habitats of Sardinia (Italy)

Background: Arbutus unedo L. is a wild tree of Mediterranean regions used as food and in traditional medicine and important for afforestation programs. There is no detailed information available on the variation of A. unedo leaves metabolome across the seasons. The leaves were analyzed by Proton nuclear magnetic resonance (1&nbsp;H NMR)-based metabolomics, comparing samples harvested across the seasons and in ten different natural habitats of Sardinia (Italy). Results: Multivariate analysis showed the impact of seasonal variation on the metabolome: glucose and quinic acid increased in summer, while in spring sucrose was accumulated. β-Arbutin, the main known active principle of A. unedo, generally reached the highest concentration in autumn. In winter, O-β-methylglucose, γ-aminobutyric acid (GABA), flavonols (quercetin-3-O-α-rhamnoside, myricetin-3-O-α-rhamnoside, kaempferol-3-O-α-rhamnoside), catechin, and gallocatechin increased. Characteristic metabolomic features were found also for samples collected in different locations. For instance, trees growing at the highest altitude and exposed to lower temperatures produced less flavonols and catechins. The only sample collected on trees growing on limestones, dolomites, and dolomitic limestones type of soil showed generally the highest content of arbutin. The highest phenolics content was found during spring, while samples collected on flowering branches in winter were the ones with the highest flavonoid content. The antioxidant activity was also variated, ranging from 1.3 to 10.1&nbsp;mg of Trolox equivalents (TE)/mL of extract, and it was positively correlated to both total phenolics and flavonoid content. Winter samples showed the lowest antibacterial activity, while summer and autumn ones exhibited the highest activity (IC50 values ranging from 17.3 to 42.3&nbsp;µg/mL against Staphylococcal species). Conclusion: This work provides 1&nbsp;H-NMR fingerprinting of A. unedo leaves, elucidating the main metabolites and their variations during seasons. On the basis of arbutin content, autumn could be considered the balsamic period of this taxon. Samples collected in this season were also the most active ones as antibacterial. Moreover, an interesting metabolomic profile enriched in catechins and flavonols was observed in leaves collected in winter on flowering branches which were endowed with high antioxidant potential

Nova karakterizacija kubičnih hamiltonskih grafov s pomočjo prirejenih kvartičnih grafov

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian ▫$I$▫-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian ▫$I$▫-graphs follows from the fact that one can choose a 1-factor in any ▫$I$▫-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected ▫$I$▫-graphs as a subfamily.Podamo potreben in zadosten pogoj za to, da je kubični graf hamiltonski, tako da analiziramo Eulerjeve obhode v določenih vpetih poddrevesih kvartičnega grafa, prirejenega kubičnemu grafu s kontrakcijo 1-factorja. Ta korespondenca je zelo koristna v primeru, ko inducira modro in rdečo 2-factorizacijo prirejenega kvartičnega grafa. Ta pogoj uporabimo za to, da karakteriziramo hamiltonske ▫$I$▫-grafe, ki so nadaljnja posplošitev posplošenih Petersenovih grafov. Karakterizacija hamiltonskih ▫$I$▫-grafov sledi iz dejstva, da lahko v kateremkoli ▫$I$▫-grafu izberemo 1-faktor na tak način, da je ustrezni prirejeni kvartični graf grafovski sveženj, ki ima za bazni graf cikličen graf, vlakno in fundamentalna faktorizacija grafovih svežnjev pa igra vlogo modre in rdeče faktorizacije. Tehnike, ki jih razvijemo, nam omogočajo predstaviti Cayleyjeve multigrafe stopnje 4, ki so pridruženi abelskim grupam, kot grafovske svežnje. Še več, najdemo lahko družino povezanih kubičnih (multi)grafov, ki vsebuje družino povezanih ▫$I$▫-grafov kot svojo poddružin

Octahedral, dicyclic and special linear solutions of some Hamilton-Waterloo problems

We give a sharply-vertex-transitive solution of each of the nine Hamilton-Waterloo problems left open by Danziger, Quattrocchi and Stevens