Discrete MathematicsCharles H.C. Little, The parity of the number of 1-factors of a graph, Discrete Math. 2 (1972), 179-181.C. H. C. Little. The parity of the number of 1-factors of a graph. Discrete Mathematics 2 (1972), 179-181....
Proof: We prove part (i) only, and leave part (ii) as an exercise. Let be the supremal value of the quantity (1) given the constraints in Problem 3, and let be the infimal value of . We need to show that . We first establish the easy inequality . If the sequence obeys the ...
Proof Choose n:=⌈k2⌉+k−1. The vertex d⌈k2⌉ is the first of the vertices d1,…,dn to be connected to the vertices A:={aj:j⩽k}. The k−1 vertices di, i=⌈k2⌉+1,…,⌈k2⌉+k−1 are connected to each vertex of A as well. Neither the vertices of ...
Hence, our work provides the first proof that it is possible to manipulate and tailor MWP systems by exploiting non-Hermitian degeneracy phase transitions, paving the way for a new class of PT-symmetric MWP applications in the generation, processing, control and distribution of microwave and ...
ProofFor an MET-LDPC code ensemble of three edge types of\({{{\mathcal{E}}}_{1}=\{1\}\),\({{{\mathcal{E}}}_{2}=\{2\}\), and\({{{\mathcal{E}}}_{12}^{c}=\{3\}\)withgi,2 = 0 for all\(i\in {C}_{1}^{c}\), the ensemble in Eq. (1) can be rew...
Proof Since R is an\(\alpha \)-region, there is an\(\alpha \)-strategy such that, for all -strategies , with , and positions , the play induced by the two strategies is either winning for\(\alpha \)or exits from R passing through a position of the escape set ...
Proof The proof is based on how the attractor computes the region. This property is trivially true for the initial set of vertices with priority p. We consider again two cases: (a) When attracting a single vertex v, v is either an \(\alpha \)-vertex with a strategy to play to Z, ...
DE WERRA, Perfectly orderable graphs are quasi-parity graphs: A short proof, Discrete Math. 68 (1988) 11l-l 13.A. Hertz and D. de Werra. Perfectly orderable graphs are quasi-parity graphs : a short proof. Discrete Math., 68 :111-113, 1988....
J. Czap, S. Jendroľ, M. Voigt, Parity vertex colouring of plane graphs, Discrete Math. 311 (6) (2011) 512-520.Parity vertex colouring of plane graphs - Czap, Jendrol’, et al. - 2011 () Citation Context ...oloring with at most 97 colors. The proof is given in Section 2....
Perfectly orderable graphs are quasi-parity graphs: a short proof Discrete Math., 68 (1988), pp. 111-113 View in ScopusGoogle Scholar [25] C.T. Hoàng Alternating orientation and alternating colouration of perfect graphs J. Combin. Theory Ser. B, 42 (1987), pp. 264-273 View in Scopus...