By Qinglin Roger Yu
"Graph components and Matching Extensions" bargains with very important branches of graph idea – issue conception and extendable graphs. as a result of mature ideas and huge levels of purposes, components and matchings turn into worthy instruments in research of many theoretical difficulties and sensible matters. This ebook contains uncomplicated innovations, theoretical foundations, in addition to contemporary advances within the box. It additionally discusses open difficulties and conjectures to motivate the readers.
This ebook is basically aimed toward researchers and graduate scholars in graph conception and combinatorics regularly. although, all of the priceless necessities are built from scratch and may be available to upper-level undergraduate scholars with previous wisdom of graph theory.
Dr. Yu is a professor within the division of arithmetic and information on the Thompson Rivers collage of Canada. His examine pursuits contain graph idea and combinatorial optimization.
Dr. Liu is a professor within the institution of arithmetic at Shandong college of China. Her learn pursuits contain graph idea and matroid theory.
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Extra resources for Graph factors and matching extensions
2 (I-Factor Theorem). Let G be a gmph and f : V(G) --+ Z+ be a flLnction. Then G has an f -factor if and only if, for all S, l' c:;; V (G) with S n l' = 0. ;) == 1 (mod 2). MOT'(xJUer. rEV(G) f(:r) (mod 2). 1. The original proof of f- Factor Theorem is lengthy and dif~ ficulty. However. there is shorter proof provided by Tutte [G6:3], who constructed a new graph G 8 from G and then proved that G has an f-factor if and only if G* has a I-factor. vYe describe this reducLion meLhod below. Assume that f (:r) ~ d( x) for all x E V (G), otherwise G has too few edges at :c to have an f-factor.
Then F - e v is a maxirrnlln matching of G - v. lVloreover, F - e v saturates all vertices of A(G - v) and has at most one edge joining each odd cOYllponent of JJ( G - v) (of which H is one) to A(G-v). Rccall S = A(G-v)Uv, upon rcinstalling v and Eel' wc havc cxactly one edge of F joining S to each component of G - S and in particular, exactly onc cdgc of F joining H to S. So E(H) n F is a ncar-pcrfcct matching of H. Thus E( II - V 2 ) n F is a near- perfect matching of II - V2 , and so IV (II) n Vii is odd.
C m be the odd components of G - 5, where Tn = o(G - 5). 1, we have cc;(Ci : 5) 2 r. Hence TI51 2 1\7(5)1 2 cc;(Cl U C2 U··· U C m : 5) 2 TTn 2 r(151 + 1) a contradiction. Hcncc G is factor-criLical. : thcrc cxisL infiniLely many (r - 2)-cdgc-conncctcd T-rcgular graphs of cvcn order that have no I-factors. The ahove theorem holds for multigraphs as well. K ote that in part (b) T mlUit be an even integer. A classic result due to Plesnik (1974) showed that there exist I-factors to avoid a set of edges in regular graphs with edge-connectivity condition.