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...<math>a_{i+1} = b_i</math> for all <math> 1 \leq i \leq k-1 </math>. Shift graphs are [[Triangle-free graph|triangle-free]], and for fixed <math>k</math> the | contribution = On chromatic number of infinite graphs ...

'''''The Strange Logic of Random Graphs''''' is a book on [[zero-one law]]s for [[random graph]]s. It was written b ...ction. A zero-one law is a theorem stating that, for certain properties of graphs, and for certain choices of <math>p</math>, ...

...ism|non-isomorphic]] [[simple graph|simple]] [[connected graph|connected]] graphs such that the [[automorphism group]] of each of them is [[group isomorphism ==Special families of graphs== ...

...tics|Does forbidding both a tree and a clique as induced subgraphs produce graphs of bounded chromatic number?}} ...nt number of colors. Equivalently, it asks whether the <math>T</math>-free graphs are [[χ-bounded|<math>\chi</math>-bounded]].{{r|s97}} ...

{{Short description|Two 3-regular graphs with 18 vertices and 27 edges}} ...those two infinite families.<ref>Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 276 and 280, 1998.</ref> ...

In [[graph theory]], the '''McKay–Miller–Širáň graphs''' are an infinite class of [[vertex-transitive graph]]s with [[Diameter (graph theory)|diamet ...l number of vertices can be at most <math>d^2+1</math>. However, only four graphs are known to reach this bound: a single edge (degree one), a 5-vertex [[cyc ...

.... Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8&thinsp;×&thinsp;8 square grid". ...of graphs with some regular structure, such as the [[Cartesian product of graphs|Cartesian product]] of a number of [[complete graph]]s.<ref name= weiss-lg> ...

The individual graphs may be named after the associated solid: ...are isomorphic to the prism graphs, and do not form a separate sequence of graphs. ...

...mplete bipartite graph]] on the same vertices. The name was given to these graphs by [[Paul Erdős]] and [[András Hajnal]].{{r|e}} The same concept can also be defined in the same way for infinite graphs over two copies of any ordered set of vertices.{{r|e}} ...

...ting that every infinite graph contains either a [[Countable set|countably infinite]] [[independent set (graph theory)|independent set]], or a [[clique (graph ...o the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order.{{r|dusmil}} ...

...}} A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic.{{r|br}} ...n efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases.{{r|zn}} ...

...lyhedron]]. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the [[k-vertex-connected graph|3-vertex-connected]], [[planar graph]]s. ...ze]] the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs. That is, whenever a graph is both planar and 3-vertex-connected, there exi ...

'''Configuration graphs''' are a theoretical tool used in [[computational complexity theory]] to pr A cycle in the graph corresponds to an infinite loop in the computation. ...

...[[longest path problem|longest paths]] and [[graph coloring|colorings]] in graphs, and to the [[Erdős–Szekeres theorem]] on monotonic subsequences. ...heorem]] of {{harvtxt|Lovász|1972}} states that the complements of perfect graphs are always perfect, and can be used to deduce Dilworth's theorem from Mirsk ...

...ion of some results, such as the [[max-flow min-cut theorem]], to infinite graphs.{{sfnp|Folkman|Fulkerson|1970}} ...ssue=1|year=1970|doi=10.1016/S0021-9800(70)80006-0|title=Flows in Infinite Graphs|last=Folkman|first=Jon|author-link=Jon Folkman|last2=Fulkerson|first2=D.R.| ...

...w-free]]. It plays an important role in the structure theory for claw-free graphs by {{harvtxt|Chudnovsky|Seymour|2005}}. ...v'' belongs uniquely to a subgraph in the form of a [[Cartesian product of graphs|Cartesian product]] of [[complete graph]]s ''K''₆ <math>\square< ...

The only [[cycle graph]]s that can be realized as zero-product graphs (with zero divisors as vertices) are the cycles of length 3 or 4.{{r|al}} ...o-divisor graphs are the [[star (graph theory)|stars]] (complete bipartite graphs that are trees) and the five-vertex tree formed as the zero-divisor graph o ...

...{r|t90|t81}} If true, it would be best possible: there are infinitely many graphs for which <math>\tau(G)=2\nu(G)</math>, including all of the [[block graph] ...(Planar graphs have degeneracy at most five.) It is also known to hold for graphs of [[treewidth]] at most six,{{r|bfg}} for [[threshold graph]]s,{{r|bbg}} f ...

...|graph]]s, referred to as the ''upper bunk'' and the ''lower bunk''. These graphs are [[Graph isomorphism|isomorphic]], meaning they share the same structure ...ation for the conjecture was its implication that, in a percolation on the infinite square grid, the probability of {{Math|(0, 0)}} being connected to {{Math|( ...

...hat is, it is [[Locally linear graph|locally linear]]. Finding large dense graphs with this property is one of the formulations of the [[Ruzsa–Szemerédi prob ==Related graphs== ...