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1966–67 Yugoslav First League

The 1966–67 Yugoslav First League season was the 21st season of the First Federal League, the top level association football league of SFR Yugoslavia, since its establishment in 1946. Sixteen teams contested the competition, with Sarajevo winning their first national title. At the end of the previous season Radnički Belgrade and NK Trešnjevka were relegated, they were replaced by Čelik. First League topscorer: Mustafa Hasanagić - 18 goals 1966–67 Yugoslav Second League 1966–67 Yugoslav Cup Yugoslavia Domestic Football Full Tables

Hallmark University

Hallmark University is a private university in San Antonio, Texas. It focuses on academic programs in Aeronautics, Healthcare and Information Technology; the school was authorized by the FAA to teach aircraft maintenance in 1969. In 1974, Richard Fessler became President of Hallmark and led in that capacity through 1999, his vision and commitment to quality education led the expansion of the college into numerous fields of training, including business and electronic engineering. The electronics program was launched in 1981. In 1982, Hallmark became the first private career school in the State of Texas to be awarded associate degree-granting authority; this expansion of offerings and a growing enrollment led to the addition of two new campuses which continue to serve the college today. In 1985, a second campus opened in the north central San Antonio area. By 1988, the business program was launched at the second campus. In 1995, Hallmark Aero-Tech was renamed Hallmark Institute of Aeronautics and Hallmark Institute of Technology.

By 2003, the computer networking, medical assisting, business office administration programs were launched. In August 2007, Hallmark Institute became Hallmark College, in 2015 became Hallmark University, it offers degrees in healthcare administration and nursing, aeronautics and computer science. Hallmark's first registered nursing program began in 2011; the university offers associate degrees, bachelor's degrees, Master of Business Administration degrees. These programs are housed in three colleges and one school: College of Aeronautics School of Information Technology School of Business School of Healthcare Member of Career Colleges and Schools of Texas Member of the Career College Association Member of Alpha Beta Kappa, a National Honor Society in Arts and Trades Member of the Professional Aviation Maintenance Association Member of the Aviation Technical Education Council Member of Higher Education Transfer Alliance North San Antonio Chamber of Commerce Greater San Antonio Chamber of Commerce Project Management Institute Registered Education Provider Accredited by the Accrediting Commission of Career Schools and Colleges Official website

Misra & Gries edge coloring algorithm

The Misra & Gries edge coloring algorithm is a polynomial time algorithm in graph theory that finds an edge coloring of any graph. The coloring produces uses at most Δ + 1 colors; this is optimal for some graphs, by Vizing's theorem it uses at most one color more than the optimal for all others. It was first published by Jayadev Misra and David Gries in 1992, it is a simplification of a prior algorithm by Béla Bollobás. This algorithm is the fastest known almost-optimal algorithm for edge coloring, executing in O time. A faster time bound of O was claimed in a 1985 technical report by Gabow et al. but this has never been published. In general, optimal edge coloring is NP-complete, so it is unlikely that a polynomial time algorithm exists. There are however exponential time exact edge coloring algorithms. A color x is said to be free of an edge on u if c ≠ x for all ∈ E: z≠v. A fan of a vertex u is a sequence of vertices F that satisfies the following conditions: F is a non-empty sequence of distinct neighbors of u ∈ E is uncolored The color of is free on F for 1 ≤ i < k Given a fan F, any edge for 1 ≤ i ≤ k is a fan edge.

Let c and d be colors. A cdX-path is an edge path that goes through vertex X, only contains edges colored c and d and is maximal. Note that only one such path exists for a vertex X, as at most one edge of each color can be adjacent to a given vertex. Given a fan F of a vertex X, the "rotate fan" operation does the following: c=c Uncolor This operation leaves the coloring valid, as for each i, c was free on; the operation "invert the cdX-path" switches every edge on the path colored c to d and every edge colored d to c. Inverting a path can be useful to free a color on X if X is one of the endpoints of the path: if X was adjacent to color c but not d, it will now be adjacent to color d, not c, freeing c for another edge adjacent to X; the flipping operation will not alter the validity of the coloring since for the endpoints, only one of can be adjacent to the vertex, for other members of the path, the operation only switches the color of edges, no new color is added. Algorithm Misra & Gries edge coloring algorithm is input: A graph G. output: A proper coloring c of the edges of G.

Let U:= E while U ≠ ∅ do Let be any edge in U. Let F be a maximal fan of u starting at F = v. Let c be a color, free on u and d be a color, free on F. Invert the cdu path Let w ∈ V be such that w ∈ F, F' = is a fan and d is free on w. Rotate F' and set c = d. U:= U − end while The correctness of the algorithm is proved in three parts. First, it is shown that the inversion of the cdu path guarantees a vertex w such that w ∈ F, F' = is a fan and d is free on w. Then, it is shown that the edge coloring is proper and requires at most Δ + 1 colors. Prior to the inversion, there are two cases: The fan has no edge colored d. Since F is a maximal fan and d is free on F, this implies there is no edge with color d adjacent to u, otherwise, if there was, this edge would be after F, as d is free on F, but F was maximal, a contradiction. Thus, d is free on u, since c is free on u, the cdu path is empty and the inversion has no effect on the graph. Set w = F; the fan has one edge with color d. Let be this edge.

Note that x + 1 ≠ 1 since is uncolored. Thus, d is free on F. Also, x ≠ k since the fan has length k but there exists a F. We can now show that after the inversion, for each y ∈, the color of is free on F. Note that prior to the inversion, the color of is not c or d, since c is free on u and has color d and the coloring is valid; the inversion only affects edges that are colored d, so holds. F can either be in the cdu path or not. If it is not the inversion will not affect the set of free colors on F, d will remain free on it. We can set w = F. Otherwise, we can show that F is still a fan and d remains free on F. Since d was free on F before the inversion and F is on the path, F is an endpoint of the cdu path and c will be free on F after the inversion; the inversion will change the color of from d to c. Thus, since c is now free on F and holds, F remains a fan. D remains free on F, since F is not on the cdu path. Select w = F. In any case, the fan F' is a prefix of F, which implies F' is a fan; this can be shown by induction on the number of colored edges.

Base case: no edge is colored, this is valid. Induction step: suppose. In the current iteration, after inverting the path, d will be free on u, by the previous result, it will be free on w. Rotating F' does not compromises the validity of the coloring. Thu