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This paper shows that every 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. ...doi:10.1016/s0012-365x(96)00288-9 fatcat:3mqh7blzyjcuphyiz7aknbcrgy
Note also that those 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  ). ... 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. ...doi:10.1016/0012-365x(90)90326-d fatcat:dy2snld5y5hfvfngoxfhiocnju
Liu, 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 ...doi:10.1016/s0012-365x(97)83859-9 fatcat:lyk7pw4tsjgt7cisllruhse7ni
[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. ...
. ÷ 1 obtained from the 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. ...doi:10.1016/0012-365x(94)00248-h fatcat:2lmlquvnwnclbh4qyxc2qgetpa
Special techniques and ideas used to produce new 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  ). ...doi:10.1016/s0012-365x(96)00269-5 fatcat:irfp7ha4frcgxowntl2jr3ojsi
In some instances the best possible bounds on both the 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. ...doi:10.1016/j.disc.2010.04.019 fatcat:nwylep7wfjfhvgb3yjia3lhhge
Journal of Kufa for Mathematics and Computer
This paper show that every 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. : 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. ...fatcat:y6l3jxzb6jdgfhecq6go5ei5ti
Liu, 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 ...doi:10.1016/s0012-365x(98)90328-4 fatcat:s2tsivncvfcilf6jlbl4fx24zq
We prove that the class of planar graphs has no finite duality except for 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. ...doi:10.1016/j.jctb.2011.06.001 fatcat:qxftndmv5jclthwoccyeaaoyv4
Many results pertaining to other more studied cases 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 ...doi:10.37236/21 fatcat:xsh3n7wd2fbibgxla3okg5dfxq
Let d min (G) (d max (G)) denote the minimum (maximum) of the number of dependent arcs over 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. ...doi:10.1016/j.ejc.2009.03.024 fatcat:c3rftnlf2fhatdpcje6e3kusdq
In addition to being among the most fascinating purely combinatorial problems, they 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. ...doi:10.1016/j.cosrev.2007.07.002 fatcat:artb4tejljclrdaq36qz4k5utq
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.) ...
This leads to 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. ...arXiv:1810.06220v1 fatcat:5nsno524m5anfdbvq4knjwegji
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