On small graphs critical with respect to edge colourings

Beineke, Lowell W.; Fiorini, Stanley

doi:10.1016/0012-365x(76)90139-4

Cited by 43 publications

(23 citation statements)

References 5 publications

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“…In view of (2) If G' satisfies (7) then G contains a near 1-factor F including av^ and avoiding b. Thus in all cases it follows that either G is Class 1 (a contradiction) or G contains a near 1-factor F including av 1 Therefore |F((?g) U {a}| > |F(<? })|-3 = 2 m -2 .…”

Section: Let the Excess E(g) Of G Be Defined By E(g) = 3a{g)-2\v(o)\mentioning

confidence: 94%

Star multigraphs with three vertices of maximum degree

Chetwynd

Hilton

1986

Math. Proc. Camb. Phil. Soc.

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The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

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Section: Let the Excess E(g) Of G Be Defined By E(g) = 3a{g)-2\v(o)\mentioning

confidence: 94%

Star multigraphs with three vertices of maximum degree

Chetwynd

Hilton

1986

Math. Proc. Camb. Phil. Soc.

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“…Let x be a j -vertex that is adjacent to a k-vertex y. If j < Δ, k < Δ, then x is adjacent to at least Δ − k + 1 vertices z satisfying the following: z = y; z is adjacent to at least 2Δ − j − k vertices different from x of degree at least 2Δ − j − k + 2; and if z is not adjacent to y, then z is adjacent to at least The next lemma summarizes the results in [3][4][5][6][7]. Proof.…”

Section: Claimmentioning

confidence: 82%

“…If d 7 (x) = 0 and d 8 (x) 1, then M (x) 1. Otherwise M (x) 3 2 . If x is adjacent to a 6-vertex, then by Lemma 2.2, each 9-neighbor of x is adjacent to at most two ( 8)-vertices.…”

Section: Lemma 28 (See Sanders and Zhaomentioning

confidence: 97%

“…By VAL, each 9-neighbor of a 5-vertex is adjacent to at most four ( 8)-vertices. Since n 5 = 0, one can find at least four 9-vertices v 1 , v 2 , v 3 , v 4 such that d 5 (v i ) 1 and d 8 (v i ) 4 for i ∈ {1, 2, 3, 4}. Hence 4(|V (G)| − n 9 ) = 4(n 5 + n 6 + n 7 + n 8 ) |[V 8 , V 9 ]| = |[V 9 , V 8 ]| 4 × 4 + 5 × (n 9 − 4) = 5n 9 − 4.…”

Section: Lemma 28 (See Sanders and Zhaomentioning

confidence: 99%

“…Similarly, we can prove that M (x)3 2 if x is adjacent to three 8-vertices.2 By Eq. (5), (II), (V), (VII) and Claims 3 and 6Claim 7. δ = 4.…”

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confidence: 93%

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Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic ϵ∈{−1,−2,−3}

Luo

Zhao

2008

Journal of Combinatorial Theory, Series B

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In this paper, we consider the problem of determining the maximum of the set of maximum degrees of class two graphs that can be embedded in a surface. For each surface Σ, we define Δ(Σ) = max{Δ(G)| G is a class two graph of maximum degree Δ that can be embedded in Σ}. Hence Vizing's Planar Graph Conjecture can be restated as Δ(Σ) = 5 if Σ is a plane. We show that Δ(Σ) = 7 if (Σ) = −1 and Δ(Σ) = 8 if (Σ) ∈ {−2, −3}.

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