Cliques in the union of graphs

Let B and R be two simple graphs with vertex set V, and let G(B, R) be the simple graph with vertex set V, in which two vertices are adjacent if they are adjacent in at least one of Band R. For X⊆V, we denote by B|X the subgraph of B induced by X; let R|X and G(B, R)|X be defined similarly. A clique in a graph is a set of pairwise adjacent vertices. A subset U⊆V is obedient if U is the union of a clique of B and a clique of R. Our first result is that if B has no induced cycles of length four, and R has no induced cycles of length four or five, then every clique of G(B, R)is obedient. This strengthens a previous result of the second author, stating the same when B has no induced C_4 and R is chordal. The clique number of a graph is the size of its maximum clique. We say that the pair (B, R)is additive if for every X⊆V, the sum of the clique numbers of B|X and R|X is at least the clique number of G(B, R)|X. Our second result is a sufficient condition for additivity of pairs of graphs

Red-blue clique partitions and (1-1)-transversals

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced $C_4$ and $C_5$ then the vertices of $K$ can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic $C_4$ and there is no induced $C_5$ in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced $C_4$ and there is no $K_5$ on which both color classes induce a $C_5$. We also answer a question of Kaiser and Rabinovich, giving an example of six $2$-convex sets in the plane such that any three intersect but there is no $(1-1)$-transversal for them

A Characterisation of Strong Integer Additive Set-Indexers of Graphs

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexers. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. We already have some characteristics of the graphs which admit strong integer additive set-indexers. In this paper, we study the characteristics of certain graph classes, graph operations and graph products that admit strong integer additive set-indexers.Comment: 10 page