If v < sub > n </ sub > is the number of vertices of degree n and D is the maximum degree of any vertex,

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

If there is an edge from a vertex x to a vertex y, then the element is 1 ( or in general the number of xy edges ), otherwise it is 0.

If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other.

If the polar vertex is " pulled down " 90 °, so that the vertex is on the equator, or transverse diameter, then it becomes the transverse graticule, upon which all spherical trigonometry is ultimately based ( if the longitudinal vertex is between the poles and equator, then it is considered an oblique graticule ).

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

If loops are permitted, then the graph with a single vertex and one loop is terminal.

If at any step a vertex has ( visited ) neighbours with the same label as itself, then the graph is not bipartite.

If n is odd each axis of symmetry connects the midpoint of one side to the opposite vertex.

For example, given a graph G =( V, E ), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p < sub > 1 </ sub > from u to w and p < sub > 2 </ sub > from w to v are indeed the shortest paths between the corresponding vertices ( by the simple cut-and-paste argument described in CLRS ).

If G is a connected graph with infinitely many vertices such that every vertex has finite degree ( that is, each vertex is adjacent to only finitely many other vertices ) then G contains an infinitely long simple path, that is, a path with no repeated vertices.

If one angle is a right angle, the orthocenter coincides with the vertex of the right angle.

If I needed a bit of colour, I'd add it in the vertex.

If there are no negative-weight cycles, then every shortest path visits each vertex at most once, so at step 3 no further improvements can be made.

If, when a vertex is colored, there exists an edge connecting it to a previously-colored vertex with the same color, then this edge together with the paths in the breadth-first search tree connecting its two endpoints to their lowest common ancestor forms an odd cycle.

If the edges of a graph are thought of as lines drawn from one vertex to another ( as they are usually depicted in illustrations ), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

If you tell him I made a pass at you he might think you misunderstood something I said or did, so instead of just telling him I made a pass, say I tried to date you and that you agreed so you could prove to him what a louse I really am.

If we look at recent art we find it preoccupied with form.

If living Jews were unavailable for study, the Bible was at hand.

If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

If nothing else, at least two good songs came out of the project, `` Out Of This World '' and `` June Comes Around Every Year ''.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If she were not at home, Mama would see to it that a fresh white rose was there.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If it proclaims that the best is yet to be, it always arouses, at least in the young, either a suspicious question or perhaps the exclamation of the Negro youth who saw on a tombstone the inscription, `` I am not dead but sleeping ''.

If a broader Atlantic community is to be formed -- and my own judgment is that it lies within the realm of both our needs and our capacity -- a ready nucleus of machinery is at hand in the NATO alliance.

If you're lyin' out in the hammock at night, and it gets kinda cool -- you know -- you just take these sides with the fringe on -- see -- and wrap 'em right over you.

Mr. Richard Preston, executive director of the New Hampshire State Planning and Development Commission, in his remarks to the Governors Conference on Industrial Development at Providence on October 8, 1960, warned against the fallacy of attempting to attract industry solely to reduce the tax rate or to underwrite municipal services such as schools when he said: `` If this is the fundamental reason for a community's interest or if this is the basic approach, success if any will be difficult to obtain ''.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

If Joe doesn't show up, we'll all be back here at 0600 hours.

If he had any worries, it was only the small ones, about Mother in New York, and his daughter Edwina and what she might be doing at this hour, with her Aunt Asia, in Philadelphia.

`` If William agrees, we should insist on a public debate '', he said at length.

If we sit here reading editorials and looking at public-opinion polls and other reports that cross our desks, we should realize that this is raw, undigested opinion expressed in the absence of leadership.

If our national interest lies in being able to fight and win a war rather than committing national suicide, then we must take a much more penetrating look at ballistic missiles.

If our SAC bombers were, today, capable of surviving a surprise missile attack and because of infinite dispersion or long endurance had the capability to strike at Russia again, and again, and again, those bombers would unquestionably assure our military dominance.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If the lever is pulled to clear a signal or move a switch, the tappet moves a short distance lengthwise at the same time.

If the background of design is too smooth, or you wish to create a wood-grained effect, it may be added at this time with a dull tool such as the handle of a fine paintbrush.