The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M1 and M2 such that (i) M1 ∪ M2 in...

We study the stability number, chromatic number and clique cover of graphs with no induced P5 and diamonds. In particular, we provide a way to obtain all imperfect (P5, diamond)-free graphs by iterated point multiplication or substitution from a /nite collection of small basic graphs. Corollaries of this and other structural properties, among which a result of Bacs1 o and Tuza, are (i) combinat...

Given a clustered graph (G,V), that is, a graph G = (V,E) together with a partition V of its vertex set, the selective coloring problem consists in choosing one vertex per cluster such that the chromatic number of the subgraph induced by the chosen vertices is minimum. This problem can be formulated as a covering problem with a 0-1 matrix M(G,V). Nevertheless, we observe that, given (G,V), it i...

We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order n is equivalent to a proper edge-coloring of Kn,n. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined l(n) as the least integer such that every properly edge-colored Kn,n, which contains at least l(n) different colors, admits a ...

Let G = (V,E) be an edge-colored graph, i.e., G is assigned a surjective function C : E → {1, 2, · · · , r}, the set of colors. A matching of G is called heterochromatic if its any two edges have different colors. Let (B,C) be an edge-colored bipartite graph and d(v) be color degree of a vertex v. We show that if d(v) ≥ k for every vertex v of B, then B has a heterochromatic matching of cardina...

Let HPn,m,k be drawn uniformly from all m-edge, k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HP (κ) n,m,k be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ = n and m = Kn log n where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in...

The upper chromatic number χ(H) of a hypergraph H is the maximum number of colors in a coloring avoiding a polychromatic edge. The stability number α(H) of a hypergraph H is the cardinality of the largest set of vertices of H which does not contain an edge. A hypergraph is k-uniform if the sizes of all its edges are k. A hypergraph H is co-perfect if χ(H ′) = α(H ′) for each induced subhypergra...

Consider a graph $G$ with coloring of its edge set $E(G)$ from $Q = \{c_1,c_2, \ldots, c_q\}$. Let $Q_i$ be the all edges colored $c_i$. Recently, Frieze defined notion perfect matching color profile denoted by $\mathrm{mcp}(G)$, which is vectors $(m_1, m_2, m_q)$ such that there exists $M$ in $|Q_i \cap M| m_i$ for $i$. $\alpha_1, \alpha_2, \alpha_q$ positive constants $\sum_{i=1}^q \alpha_i 1...

Given a graph G = (V,E) and a “cost function” f : 2 → R (provided by an oracle), the problem [PCliqW] consists in finding a partition into cliques of V (G) of minimum cost. Here, the cost of a partition is the sum of the costs of the cliques in the partition. We provide a polynomial time dynamic program for the case where G is an interval graph and f belongs to a subclass of submodular set func...

When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinatorial group-theoretic invariants associated to the boundaries of the tile shapes and the regions t...