A graph G is hypohamiltonian if G is non-hamiltonian and G − v is hamiltonian for every v ∈ V (G). In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 gra...

The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M1 and M2 such that (i) M1 ∪ M2 in...

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, are assigned different colors. The strong chromatic index of G is the smallest number k for which G has a strong k-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree tw...

a graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. in this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

In 1996, Reed proved that the domination number γ (G) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8. This bound is sharp for cubic graphs if there is no restriction on connectivity. In this paper we show that γ (G) ≤ 4n/11 for every n-vertex cubic connected graph G if n > 8. Note that Reed’s conjecture that γ (G) ≤ dn/3e for every connected cubic n-vertex graph G is n...

In 1996, Reed proved that the domination number (G) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8.Also, he conjectured that (H) n/3 for every connected 3-regular (cubic) n-vertex graph H. In this note, we disprove this conjecture. We construct a connected cubic graph G on 60 vertices with (G) = 21 and present a sequence {Gk}∞k=1 of connected cubic graphs with lim k→∞ ...

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 sim...

A graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. In this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

A paired-dominating set of a graph is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph. Recently, Chen, Sun and Xing [Acta Mathematica Scientia Series A Chinese Edition 27(1) (2007), 166–170] proved that a cubic graph has paired-domination number at most three-fifths ...

By a symmetric graph we mean a graph X which automorphism group acts transitively on the arcs of X. A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. Tutte [31, 32] showed that every finite symmetric cubic graph is s-regular for some s ≤ 5. It is well-known that there are precisely five symmetric cubic graphs of girth less than 6. All these graphs can be re...