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.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

Vertex partitions of r -edge-colored graphs

Let G be an edge-colored graph. The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G. In the authors’ previous work, it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P = NP. In this paper the authors show that for any fixed integer r ≥ 5, if the edges of a graph G are colored by r colors, called an r-edge-colored graph, the problem remains NP-complete. Similar result holds for the monochromatic path (cycle) partition problem. Therefore, to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given. Supported by the National Natural Science Foundation of China, PCSIRT and the “973” Program.     

Large dense neighbourhoods and turán's theorem

Let t_r(n) denote the number of edges in the complete r-partite graph on n vertices whose colour classes are as equal in size as possible. It is proved that if G is a simple graph on n vertices and more than t_r(n) edges, and if v is any vertex of maximum degree m in G, then the subgraph of G induced by the neighbours of v has more than t_r?1(m) edges. Examples are given to demonstrate the sharpness of this theorem, and its algorithmic implications are noted.     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t_μ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(₂^P)?p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(₂^P)?p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t₀(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.     

Some Algebraic Methods for Calculating the Number of Colorings of a Graph

With an arbitrary graph G having n vertices and m edges, and with an arbitrary natural number p, we associate in a natural way a polynomial R(x ₁,...,x _n) with integer coefficients such that the number of colorings of the vertices of the graph G in p colors is equal to p ^m-n R(0,...,0). Also with an arbitrary maximal planar graph G, we associate several polynomials with integer coefficients such that the number of colorings of the edges of the graph G in 3 colors can be calculated in several ways via the coefficients of each of these polynomials. Bibliography: 2 titles.     

[r,s, t]-Colorings of Graph Products

Let G =  (V, E) be a graph with vertex set V and edge set E. Given non negative integers r, s and t, an [r, s, t]-coloring of a graph G is a proper total coloring where the neighboring elements of G (vertices and edges) receive colors with a certain difference r between colors of adjacent vertices, a difference s between colors of adjacent edges and a difference t between colors of a vertex and an incident edge. Thus [r, s, t]-colorings generalize the classical colorings of graphs and can have applications in different fields like scheduling, channel assignment problem, etc. The [r, s, t]-chromatic number χ _r,s,t (G) of G is the minimum k such that G admits an [r, s, t]-coloring. In our paper we propose several bounds for the [r, s, t]-chromatic number of the cartesian and direct products of some graphs.     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Bipartite rainbow numbers of matchings

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number ofHwith respect toG, and simply called the bipartite rainbow number ofH if G is the complete bipartite graph K_m,n. Erd?s, Simonovits and Sós showed that rb(K_n,K₃)=n. In 2004, Schiermeyer determined the rainbow numbers rb(K_n,K_k) for all n≥k≥4, and the rainbow numbers rb(K_n,kK₂) for all k≥2 and n≥3k+3. In this paper we will determine the rainbow numbers rb(K_m,n,kK₂) for all k≥1.     

Total chromatic number of one kind of join graphs

The total chromatic number χ_T(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if χ_T(G)=Δ(G)+1. In this paper we prove that the join of a complete inequibipartite graph K_n1,n2 and a path P_m is of Type 1.     

Sum List Coloring Graphs

Let G=(V,E) be a graph with n vertices and e edges. The sum choice number of G is the smallest integer p such that there exist list sizes (f(v):v ∈ V) whose sum is p for which G has a proper coloring no matter which color lists of size f(v) are assigned to the vertices v. The sum choice number is bounded above by n+e. If the sum choice number of G equals n+e, then G is sum choice greedy. Complete graphs K_n are sum choice greedy as are trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocks is sum choice greedy is also sum choice greedy. We also determine the sum choice number of K_2,n, and we show that every tree on n vertices can be obtained from K_n by consecutively deleting single edges where all intermediate graphs are sc-greedy.     

Double cycle covers and the petersen graph

Let O(G) denote the set of odd-degree vertices of a graph G. Let t ? N and let ??_t denote the family of graphs G whose edge set has a partition. E(g) = E₁ U E₂ U … U E_tsuch that O(G) = O(G[E_i]) (1 ? i ? t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ? ??₃, G has at least one cut-edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ? ??_2p+1(p ? N).     

A degree bound on decomposable trees

A n-vertex graph is said to be decomposable if for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.     

On the number of edge disjoint cliques in graphs of given size

In this paper, we prove that any graph ofn vertices andt _r–1(n)+m edges, wheret _r–1(n) is the Turán number, contains (1–o(1)m edge disjointK _r'sifm=o(n ²). Furthermore, we determine the maximumm such that every graph ofn vertices andt _r–1(n)+m edges containsm edge disjointK _r's ifn is sufficiently large.Research partially supported by Hungarian National Foundation for Scientific Research Grant no. 1812.