quartic graph

File:Chvatal Lombardi.svg]]

Several well-known graphs are quartic. They include:

• The complete graph K₅, a quartic graph with 5 vertices, the smallest possible quartic graph.
• The Chvátal graph, another quartic graph with 12 vertices, the smallest quartic graph that both has no triangles and cannot be colored with three colors.{{citation

| last = Chvátal | first = V. | author-link = Václav Chvátal

| doi = 10.1016/S0021-9800(70)80057-6

| issue = 1

| journal = Journal of Combinatorial Theory

| pages = 93–94

| title = The smallest triangle-free 4-chromatic 4-regular graph

| volume = 9

| year = 1970| doi-access = free

}}.

• The Folkman graph, a quartic graph with 20 vertices, the smallest semi-symmetric graph.{{citation

| last = Folkman | first = Jon

| journal = Journal of Combinatorial Theory

| mr = 0224498

| pages = 215–232

| title = Regular line-symmetric graphs

| volume = 3

| year = 1967

| doi=10.1016/s0021-9800(67)80069-3| doi-access = free

}}.

• The Meredith graph, a quartic graph with 70 vertices that is 4-connected but has no Hamiltonian cycle, disproving a conjecture of Crispin Nash-Williams.{{citation

| last = Meredith | first = G. H. J.

| journal = Journal of Combinatorial Theory

| mr = 0311503

| pages = 55–60

| series = Series B

| title = Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs

| volume = 14

| year = 1973

| doi=10.1016/s0095-8956(73)80006-1| doi-access = free

}}.

Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs.{{citation

| last1 = Bondy | first1 = J. A.

| last2 = Häggkvist | first2 = R.

| doi = 10.1007/BF02190157

| issue = 1

| journal = Aequationes Mathematicae

| mr = 623315

| pages = 42–45

| title = Edge-disjoint Hamilton cycles in 4-regular planar graphs

| volume = 22

| year = 1981}}. Knot diagrams and link diagrams are also quartic plane multigraphs, in which the vertices represent the crossings of the diagram and are marked with additional information concerning which of the two branches of the knot crosses the other branch at that point.{{citation

| last = Welsh | first = Dominic J. A. | authorlink = Dominic Welsh

| contribution = The complexity of knots

| doi = 10.1016/S0167-5060(08)70385-6

| location = Amsterdam

| mr = 1217989

| pages = 159–171

| publisher = North-Holland

| series = Annals of Discrete Mathematics

| title = Quo vadis, graph theory?

| volume = 55

| year = 1993}}. The line graph of any cubic graph is quartic; the cuboctahedral graph is an example, as the line graph of a cube graph.

Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour.

And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may find a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time. The same method can also be used to color the edges of the graph with four colors in linear time.{{citation

| last = Gabow | first = Harold N. | author-link = Harold N. Gabow

| issue = 4

| journal = International Journal of Computer and Information Sciences

| mr = 0422081

| pages = 345–355

| title = Using Euler partitions to edge color bipartite multigraphs

| volume = 5

| year = 1976

| doi=10.1007/bf00998632}}.

Quartic graphs have an even number of Hamiltonian decompositions.{{citation

| last = Thomason | first = A. G.

| journal = Annals of Discrete Mathematics

| mr = 499124

| pages = 259–268

| title = Hamiltonian cycles and uniquely edge colourable graphs

| volume = 3

| year = 1978

| doi=10.1016/s0167-5060(08)70511-9}}.

It is an open conjecture whether all quartic Hamiltonian graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. The answer is known to be false for quartic multigraphs.{{citation

| last = Fleischner | first = Herbert

| doi = 10.1002/jgt.3190180503

| issue = 5

| journal = Journal of Graph Theory

| mr = 1283310

| pages = 449–459

| title = Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs

| volume = 18

| year = 1994}}.

{{commons category|4-regular graphs}}

• Cubic graph

{{reflist}}

• {{mathworld|id=QuarticGraph|title=Quartic Graph}}

{{DEFAULTSORT:Quartic Graph}}

Category:Graph families

Category:Regular graphs