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On odd and semi-odd linear partitions of cubic graphs

100%

Fouquet J.-L., Thuillier H., Vanherpe J.-M., Wojda A.

Discussiones Mathematicae Graph Theory

2009

tom 29

nr 2

275-292

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition $L = (L_B,L_R)$ is said to be odd whenever each path of $L_B ∪ L_R$ has odd length and semi-odd whenever each path of $L_B$ (or each path of $L_R$) has odd length. In [2] Aldred and Wormald showed that a cubic graph G is 3-edge colourable if and only if G has an odd linear partition. We give here more precise results and we study moreover relationships between semi-odd linear partitions and perfect matchings.

On a family of cubic graphs containing the flower snarks

100%

Fouquet J.-L., Thuillier H., Vanherpe J.-M.

Discussiones Mathematicae Graph Theory

2010

tom 30

nr 2

289-314

We consider cubic graphs formed with k ≥ 2 disjoint claws $C_i ~ K_{1,3}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of $C_i$ are joined to the three vertices of degree 1 of $C_{i-1}$ and joined to the three vertices of degree 1 of $C_{i+1}$. Denote by $t_i$ the vertex of degree 3 of $C_i$ and by T the set ${t₁,t₂,...,t_{k-1}}$. In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ {1,2,3}) is the graph where the set of vertices $⋃_{i = 0}^{i = k-1} V(C_i)∖T$ induce j cycles (note that the graphs FS(2,2p+1), p ≥ 2, are the flower snarks defined by Isaacs [8]). We determine the number of perfect matchings of every FS(j,k). A cubic graph G is said to be 2-factor hamiltonian if every 2-factor of G is a hamiltonian cycle. We characterize the graphs FS(j,k) that are 2-factor hamiltonian (note that FS(1,3) is the "Triplex Graph" of Robertson, Seymour and Thomas [15]). A strong matching M in a graph G is a matching M such that there is no edge of E(G) connecting any two edges of M. A cubic graph having a perfect matching union of two strong matchings is said to be a Jaeger's graph. We characterize the graphs FS(j,k) that are Jaeger's graphs.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Fouquet J.-L.

2 Thuillier H.

2 Vanherpe J.-M.

1 Wojda A.

1 2010

1 2009

Wyniki wyszukiwania

On odd and semi-odd linear partitions of cubic graphs

On a family of cubic graphs containing the flower snarks