Generalization of matching extensions in graphs (III)

Proposing them as a general framework, Liu and Yu (2001) [6] introduced (n,k,d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n,k and d be non-negative integers such that n+2k+d+2?|V(G)| and |V(G)|−n−d is even. If on deleting any n vertices from G the remaining subgraph H of G contains a k-matching and each k-matching can be extended to a defect-d matching in H, then G is called an (n,k,d)-graph. In this paper, we obtain more properties of (n,k,d)-graphs, in particular the recursive relations of (n,k,d)-graphs for distinct parameters n,k and d. Moreover, we provide a characterization for maximal non-(n,k,d)-graphs.     

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

Preperfect graphs

We say that a vertexx of a graph is predominant if there exists another vertexy ofG such that either every maximum clique ofG containingy containsx or every maximum stable set containingx containsy. A graph is then called preperfect if every induced subgraph has a predominant vertex. We show that preperfect graphs are perfect, and that several well-known classes of perfect graphs are preperfect. We also derive a new characterization of perfect graphs.     

On packing bipartite graphs

G andH, two simple graphs, can be packed ifG is isomorphic to a subgraph of , the complement ofH. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees ofG andH. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.the paper was written while the authors were graduate students at the University of Chicago and was completed while the first author was at M.I.T. The work of the first author was supported in part by the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648. The work of the second author was supported in part by NSF grant CCR-8706518.     

RandomlyH-coverable graphs

A graph israndomly matchable if every matching of the graph is contained in a perfect matching. We generalize this notion and say that a graphG israndomly H-coverable if every set of independent subgraphs, each isomorphic toH, that does not cover the vertices ofG can be extended to a larger set of independent copies ofH. Various problems are considered for the situation whereH is a path. In particular, we characterize the graphs that are randomlyP ₃ -coverable.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

On kernels in perfect graphs

A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.     

4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to K_1,3. The kth power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are , and T_r, r?2, where T_r is the graph with 4r vertices u_i,v_i,x_i,y_i, 1?i?r, such that for each i with 1?i?r, {u_i,v_i,x_i,y_i} is a clique of T_r and . We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K₄×K₂, and .     

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

On a matroid defined by ear-decompositions of graphs

A. Frank described in [1] an algorithm to determine the minimum number of edges in a graph G whose contraction leaves a factor-critical graph and he asked if there was an algorithm for the weighted version of the problem. We prove that the minimal critical-making edge-sets form the bases of a matroid and hence the matroid greedy algorithm gives rise to the desired algorithm.Partially supported by OTKA F014919, OTKA T17181 and OTKA T17580.     

Spectral radius of graphs with given matching number

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K_n if n = 2β or 2β + 1; if 2β + 2 ? n < 3β + 2; or if n = 3β + 2; if n > 3β + 2, where is the empty graph on t vertices.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Complete multipartite decompositions of complete graphs and complete <Emphasis Type="Italic">n</Emphasis>-partite graphs

In this paper, a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q _n is proposed and another new concept of a normal complete multipartite decomposition of K _n is introduced. It is showed that an optimal complete multipartite decomposition of type 1 of K _n is a normal complete multipartite decomposition. As for any complete multipartite decomposition of K _n , there is a derived complete multipartite decomposition for Q _n . It is also showed that any optimal complete multipartite decomposition of type 1 of Q _n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K _n . Besides, some structural properties of an optimal complete multipartite decomposition of type 1 of K _n are given. Supported by the National Natural Science Foundation of China (10271110).     

Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G, t(G), is defined as t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1} where ω(G-S) denotes the number of components of G-S or t(G)=+∞ if G is a complete graph. Much work has been contributed to the relations between toughness and the existence of factors of a graph. In this paper, we consider the relationship between the toughness and the existence of fractional k-factors. It is proved that a graph G has a fractional 1-factor if t(G)?1 and has a fractional k-factor if t(G)?k-1/k where k?2. Furthermore, we show that both results are best possible in some sense.     

Fractionally colouring total graphs

Bchzad and Vizing have conjectured that given any simple graph of maximum degree , one can colour its edges and vertices with +2 colours so that no two adjacent vertices, or two incident edges, or an edge and either of its ends receive the same colour. We show that for any simple graphG, V(G)E(G) can be fractionally coloured with +2 colours.     

Small cycle double covers of 4-connected planar graphs

Acycle double cover of a graph,G, is a collection of cycles,C, such that every edge ofG lies in precisely two cycles ofC. TheSmall Cycle Double Cover Conjecture, proposed by J. A. Bondy, asserts that every simple bridgeless graph onn vertices has a cycle double cover with at mostn–1 cycles, and is a strengthening of the well-knownCycle Double Cover Conjecture. In this paper, we prove Bondy's conjecture for 4-connected planar graphs.     

Heterochromatic tree partition numbers for complete bipartite graphs

An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by t_r(G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we give an explicit formula for the heterochromatic tree partition number of an r-edge-colored complete bipartite graph K_m,n.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).