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All wheels with two missing consecutive spokes are chromatically unique

1998
*
Discrete Mathematics
*

This paper shows that every

doi:10.1016/s0012-365x(96)00288-9
fatcat:3mqh7blzyjcuphyiz7aknbcrgy
*wheel*of order n, n >~ 6,*with**two**missing**consecutive**spokes*is*chromatically**unique*. (~) ... Let W~ be the*wheel*of order n and let W(n,k) be the graph obtained from W~ by deleting*all*but k*consecutive**spokes*, where n >/4 and 1 ~<k ~<n-1. ...*Two*graphs G and H*are*called*chromatically*equivalent if P(G, 2) = P(H, 2) , and a graph G is called*chromatically**unique*if P(H,2)=-P(G, 2) implies that H is isomorphic to G for any graph H. ...##
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The chromaticity of wheels with a missing spoke

1990
*
Discrete Mathematics
*

Note also that those

doi:10.1016/0012-365x(90)90326-d
fatcat:dy2snld5y5hfvfngoxfhiocnju
*wheels**with*n -3 and n -4*consecutive**missing**spokes*have been shown to be*chromatically**unique*for IZ 2 5 and n Z= 6 respectively (see [2, Theorems 2, 31). ... Here we shall show that the graph U"" obtained by deleting a*spoke*from Wn+I is*chromatically**unique*if 12 2 3 is odd. Note that U" U, and U,*are**all**chromatically**unique*while U, is not (see [2] ). ... In proving the theorem, we*are*making use of the facts that U,,, is*uniquely*3-colorable if n 3 3 is odd, and that the subgraph induced by any*two*color classes of the*unique*3-coloring is a tree. ...##
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Author index to volume 184 (1998)

1998
*
Discrete Mathematics
*

Liu,

doi:10.1016/s0012-365x(97)83859-9
fatcat:lyk7pw4tsjgt7cisllruhse7ni
*All**wheels**with**two**missing**consecutive**spokes**are*chromati-, R. and C.J. ... Huang, A study of the total*chromatic*number of equibipartite graphs (1-3) 49-60 Chen, X., Some families of*chromatically**unique*bipartite graphs (Note) , G.F., Yet another generalization of the Kruskal-Katona ...##
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Page 7556 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

[Liu, Yan Pei] (PRC-NJT; Beijing)

*All**wheels**with**two**missing**consecutive**spokes**are**chromatically**unique*. (English summary) Discrete Math. 184 (1998), no. 1-3, 71-85. ... Summary: “This paper shows that every*wheel*of order n, n > 6,*with**two**missing**consecutive**spokes*is*chromatically**unique*.” 98m:05067 05CIi5 Hilton, A. J. W. (4-RDNG; Reading); Zhao, C. ...##
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The chromaticity of wheels with a missing spoke II

1996
*
Discrete Mathematics
*

. ÷ 1 obtained from the

doi:10.1016/0012-365x(94)00248-h
fatcat:2lmlquvnwnclbh4qyxc2qgetpa
*wheel*W n ÷ 1 by deleting a*spoke*is*uniquely*determined by its*chromatic*polynomial if n >i 3 is odd. ... A graph G is*chromatically**unique*if it is*uniquely*determined by its*chromatic*polynomial P(G;2). The*wheel*IV.+ 1 is obtained by taking the join of ... .+~, is*chromatically**unique*for*all*rn t> 1 if n/> 3 is odd. I wonder if the result is still true for*all*m >t 1 and*all*even n/> 4 except n ---6. ...##
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The search for chromatically unique graphs — II

1997
*
Discrete Mathematics
*

Special techniques and ideas used to produce new

doi:10.1016/s0012-365x(96)00269-5
fatcat:irfp7ha4frcgxowntl2jr3ojsi
*chromatically**unique*graphs or*chromatically*equivalence classes*are*highlighted. ... Some of the new and relevant results on*chromatically*equivalence class*are*also included. ... Let W(n, k) denote the graph (a broken*wheel*) obtained from W~ by deleting*all*but k*consecutive**spokes*. Recall that for k ~< 5 the*chromaticities*have been decided (see [34] ). ...##
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The thickness and chromatic number of r-inflated graphs

2010
*
Discrete Mathematics
*

In some instances the best possible bounds on both the

doi:10.1016/j.disc.2010.04.019
fatcat:nwylep7wfjfhvgb3yjia3lhhge
*chromatic*number and thickness*are*achieved. We end*with*several open problems. ...*Two*plane drawings of the clone of P 3*are*given in Figure 1 . ... Let T be a tree*with*at least*two*vertices and assume that*all*trees on fewer vertices than T*are*planar when cloned. Let w be a leaf of T*with*neighbor v. ...##
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CHROMATICITY OF WHEELS WITH MISSING THREE SPOKES

2014
*
Journal of Kufa for Mathematics and Computer
*
unpublished

This paper show that every

fatcat:y6l3jxzb6jdgfhecq6go5ei5ti
*wheel*of even order*with*three*missing**spokes*is*chromatically**unique*. ... Let be*wheel*of order and let be the graph obtained from by deleting*all*but*consecutive**spokes*, where and -. ... Define to be [ ] for*all*and*with*Lemma 2.5. [5]: If and*are*forests, and or x is a tree for every , then is a chordal graph. 3. The*Chromatic**Uniqueness*of When is Even. ...##
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Master index of volumes 181–190

1998
*
Discrete Mathematics
*

Liu,

doi:10.1016/s0012-365x(98)90328-4
fatcat:s2tsivncvfcilf6jlbl4fx24zq
*All**wheels**with**two**missing**consecutive**spokes**are**chromatically**unique*184 (1998) Du, B., Kl.v2-factorization of complete bipartite graphs (Note) 187 (1998) Du, D.-Z., see F. ... Huang, A study of the total*chromatic*number of equibipartite graphs 184 (1998) Chen, C., Matchings and matching extensions in graphs 186 (1998) Chen, X., Some families of*chromatically**unique*bipartite ...##
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Finite dualities and map-critical graphs on a fixed surface

2012
*
Journal of combinatorial theory. Series B (Print)
*

We prove that the class of planar graphs has no finite duality except for

doi:10.1016/j.jctb.2011.06.001
fatcat:qxftndmv5jclthwoccyeaaoyv4
*two*trivial cases. ... Equivalently, our first result shows that for every planar core graph H (except K 1 and K 4 ) there*are*infinitely many minimal planar obstructions for H-coloring, whereas our later result gives a converse ... The edge between*two**consecutive*rims is called a rim-edge. An edge connecting a hub to its rim is called a*spoke*. ...##
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Small Ramsey Numbers

2011
*
Electronic Journal of Combinatorics
*

Many results pertaining to other more studied cases

doi:10.37236/21
fatcat:xsh3n7wd2fbibgxla3okg5dfxq
*are*also presented. We give references to*all*cited bounds and values, as well as to previous similar compilations. ... We present data which, to the best of our knowledge, includes*all*known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs*are*complete ... P i is a path on i vertices, C i is a cycle of length i , and W i is a*wheel**with*i −1*spokes*, i.e. a graph formed by some vertex x , connected to*all*vertices of the cycle C i −1 (thus W i = K 1 + C i ...##
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On preserving full orientability of graphs

2010
*
European journal of combinatorics (Print)
*

Let d min (G) (d max (G)) denote the minimum (maximum) of the number of dependent arcs over

doi:10.1016/j.ejc.2009.03.024
fatcat:c3rftnlf2fhatdpcje6e3kusdq
*all*acyclic orientations of G. ... We call G fully orientable if G has an acyclic orientation*with*exactly k dependent arcs for every k satisfying d min (G) k d max (G). ... If we attach a cycle C of type (l 1 ), l 1 = n 3, to a single vertex graph G, then G C , denoted W n , is commonly known as the*wheel**with*n*spokes*. Theorem 12. The*wheel*W n is fully orientable. ...##
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Combinatorial and computational aspects of graph packing and graph decomposition

2007
*
Computer Science Review
*

In addition to being among the most fascinating purely combinatorial problems, they

doi:10.1016/j.cosrev.2007.07.002
fatcat:artb4tejljclrdaq36qz4k5utq
*are*often motivated by algorithmic applications. ... There*are*a considerable number of intriguing fundamental problems and results in this area, and the goal of this paper is to survey the state of the art. * ... A*wheel*on n ≥ 6 vertices and n − 1*spokes*is*chromatically*2-connected although it is not complete r-partite for any r. Bipartite graphs*are*only*chromatically*1-connected. ...##
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Page 153 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

Non-chordal graphs having integral-root

*chromatic*polynomials (English summary) 98k:05064 — (*with*Liu, Yan Pei)*All**wheels**with**two**missing**consecutive**spokes**are**chromatically**unique*. ... The*chromaticity*of odd*wheels**with*a*missing**spoke*(English summary) 98f:05059 — (*with*Koh, K. M.) ...##
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Dodgson polynomial identities
[article]

2018
*
arXiv
*
pre-print

This leads to

arXiv:1810.06220v1
fatcat:5nsno524m5anfdbvq4knjwegji
*two*new identities that relate large sums of products of Dodgson polynomials to a much simpler expression involving powers of the Kirchhoff polynomial. ... In this article a new combinatorial interpretation and a generalisation of Dodgson polynomials*are*provided. ... Let G be the*wheel**with*three*spokes*depicted on the left of fig. 1. ...
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