Papers by Michael Plummer
The electronic journal of combinatorics
Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which... more Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We provide a counterexample to this conjecture.
Domination in Graphs
Structural Analysis of Complex Networks, 2010
A set of vertices S in a graph G dominates G if every vertex in G is either in S or adjacent to a... more A set of vertices S in a graph G dominates G if every vertex in G is either in S or adjacent to a vertex in S. The size of any smallest dominating set is called the domination number of G. Two variants on this concept that have attracted recent interest are total domination and connected domination. A set of
Matchings in 3-vertex-critical graphs: The even case
Networks, 2005
A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is ad... more A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)γ(G), for every
Cyclic coloration of 3-polytopes
Journal of Graph Theory, 1987
ABSTRACT A cyclic coloration of a planar graph G is an assignment of colors to the points of G su... more ABSTRACT A cyclic coloration of a planar graph G is an assignment of colors to the points of G such that for any face bounding cycle the points of F have different colors. We observe that the upper bound 2ρ*(G), due to O. Ore and M. D. Plummer, can be improved to ρ*(G) + 9 when G is 3-connected (ρ* denotes the size of a maximum face). The proof uses two principal tools: the theory of Euler contributions and recent results on contractible lines in 3-connected graphs by K. Ando, H. Enomoto and A. Saito.
Proximity thresholds for matching extension in planar and projective planar triangulations
Journal of Graph Theory, 2011
Journal of Graph Theory, 2006
Let ðGÞ be the domination number of a graph G. Reed proved that every graph G of minimum degree a... more Let ðGÞ be the domination number of a graph G. Reed proved that every graph G of minimum degree at least three satisfies ðGÞ ð3=8ÞjGj, and conjectured that a better upper bound can be obtained for cubic graphs. In this paper, we prove that a 2-edge-connected cubic graph G of girth at least 3k satisfies ðGÞ ðð3k þ 2ÞÞ=ð9k þ 3ÞjGj. For k ! 3, this gives ðGÞ ð11=30ÞjGj, which is better than Reed's bound. In order to obtain this bound, we actually prove a more general theorem for graphs with a 2-factor. ß
Journal of Combinatorial Theory, Series B, 2008
A graph G with at least 2n + 2 vertices is said to be n-extendable if every matching of size n in... more A graph G with at least 2n + 2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ , and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0, there are infinitely many graphs of orientable genus g which are 3-extendable, and given g 2, there are infinitely many graphs of non-orientable genus g which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g > 0 and sufficiently large representativity, then it is 2-extendable.
Journal of Combinatorial Theory, Series B, 2005
Since (G) · (G) n(G), Hadwiger's conjecture implies that any graph G has the complete graph K n/ ... more Since (G) · (G) n(G), Hadwiger's conjecture implies that any graph G has the complete graph K n/ as a minor, where n = n(G) is the number of vertices of G and = (G) is the maximum number of independent vertices in G. Duchet and Meyniel [Ann. Discrete Math. 13 (1982) 71-74] proved that any G has K n/(2 −1) as a minor. For (G) = 2 G has K n/3 as a minor. Paul Seymour asked if it is possible to obtain a larger constant than 1 3 for this case. To our knowledge this has not yet been achieved. Our main goal here is to show that the constant 1/(2 − 1) of Duchet and Meyniel can be improved to a larger constant, depending on , for all 3. Our method does not work for = 2 and we only present some observations on this case.
Journal of Combinatorial Theory, Series B, 1998
In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded i... more In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus #Ä >0, #Ä {2) there is a complete graph of orientable genus # (respectively, nonorientable genus #Ä ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus #Ä ), for every # (respectively #Ä ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, #Ä >0, #Ä {2) there is a complete graph K n which embeds in S # (respectively N #Ä ) such that n is the chromatic number of surface S # (respectively, the chromatic number of surface N #Ä ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity } max of the orientable surface S # to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity } gen (S # ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds } max (N #Ä ) and } gen (N #Ä ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity } max (S # ) or } max (N #Ä ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(K n )+1, #(K n+1 )] and [#Ä (K n )+1, #Ä (K n+1 )] with respect to their genus and maximum connectivities.
Bounding the Size of Equimatchable Graphs of Fixed Genus
Graphs and Combinatorics, 2009
ABSTRACT
Maximum and minimum toughness of graphs of small genus
Discrete Mathematics, 1997
A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus ... more A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus γ(G) is obtained. For γ > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ϰ(G)/2 − 1. For ϰ = 1 this bound is
Matching properties in domination critical graphs
Discrete Mathematics, 2004
A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γ... more A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γc(G) denote the size of any smallest connected dominating set in G. A graph G is k-γ-connected-critical if γc(G)=k, but if any edge e∈E(G¯) is added to G, then γc(G+e)⩽k-1. This is a variation on the earlier concept of criticality of
Two results on matching extensions with prescribed and proscribed edge sets
Discrete Mathematics, 1999
Let G be a graph with at least 2(m + n + 1) vertices. Then G is E(m,n) if for each pair of disjoi... more Let G be a graph with at least 2(m + n + 1) vertices. Then G is E(m,n) if for each pair of disjoint matchings M,N⊆E(G) of size m and n, respectively, there exists a perfect matching F in G such that M⊆F and F∩N=∅. In this paper, we prove two results concerning the property E(m,n). The first involves the
Discrete Mathematics, 2007
A graph G is said to be k--critical if the size of any minimum dominating set of vertices is k, b... more A graph G is said to be k--critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k − 1 vertices. The structure of k--critical graphs remains far from completely understood when k 3.
Discrete Mathematics, 1999
A graph G is said to be n-factor-critical if G − T has a perfect matching for each T ⊂ V (G) with... more A graph G is said to be n-factor-critical if G − T has a perfect matching for each T ⊂ V (G) with |T | = n. In this note we give a su cient condition for a graph to be n-factor-critical. Let G be a k-connected graph of order p, and let n be an integer with 06n6k and p ≡ n (mod 2) and be a real number with 1 2 6 61. We prove that if |NG(A)| ¿ (p − 2k + n − 2) + k for every independent set A of G with |A| = (k − n + 2) , then G is n-factor-critical. We also discuss the sharpness of the result and the relation with matching extension.
Discrete Mathematics, 1999
Let G be a graph with at least 2(m + n + 1) vertices. Then G is E(m,n) if for each pair of disjoi... more Let G be a graph with at least 2(m + n + 1) vertices. Then G is E(m,n) if for each pair of disjoint matchings M,N C_E(G) of size m and n, respectively, there exists a perfect matching F in G such that M C F and F n N = 0. In this paper, we extend previous results due to Chen (Discrete Math., to appear) as well as results of the present authors (Aldred et al., Discrete Math., to appear) concerning the property E(m, n). The first extends a result on claw-free graphs and the second generalizes a result about bipartite graphs. (~)
On certain spanning subgraphs of embeddings with applications to domination
Discrete Mathematics, 2009
ABSTRACT We prove the existence of certain spanning subgraphs of graphs embedded in the torus and... more ABSTRACT We prove the existence of certain spanning subgraphs of graphs embedded in the torus and the Klein bottle. Matheson and Tarjan proved that a triangulated disc with n vertices can be dominated by a set of no more than n/3 of its vertices and thus, so can any finite graph which triangulates the plane. We use our existence theorems to prove results closely allied to those of Matheson and Tarjan, but for the torus and the Klein bottle.
Matchings in 3-vertex-critical graphs: The odd case
Discrete Mathematics, 2007
ABSTRACT A subset of vertices D of a graph G is a dominating set for G if every vertex of G not i... more ABSTRACT A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.
Restricted matching in graphs of small genus
Discrete Mathematics, 2008
A graph G with at least 2n+2 vertices is said to be n-extendable if every set of n disjoint edges... more A graph G with at least 2n+2 vertices is said to be n-extendable if every set of n disjoint edges in G extends to (i.e., is a subset of) a perfect matching. More generally, a graph is said to have property E(m,n) if, for every matching M of size m and every matching N of size n in G such
Discrete Mathematics, 2001
Let G be a connected graph with at least 2(m + n + 1) vertices. Then G is E(m; n) if for each pai... more Let G be a connected graph with at least 2(m + n + 1) vertices. Then G is E(m; n) if for each pair of disjoint matchings M; N ⊆ E(G) of size m and n, respectively, there exists a perfect matching F in G such that M ⊆ F and F ∩ N = ∅. In the present paper, we wish to study the property E(m; n) for the various values of integers m and n when the graphs in question are restricted to be planar. It is known (Plummer, Annals of Discrete Mathematics 41 (1989) 347-354) that no planar graph is E(3; 0). This result is improved in the present paper by showing that no planar graph is E(2; 1). This severely limits the values of m and n for which a planar graph can have property E(m; n) and leads us to consider the properties E(1; n) and E(0; n) for certain classes of planar graphs. Sharpness of the various results is also explored.
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Papers by Michael Plummer