The irregularity strength and cost of the union of cliques

Assign positive integer weights to the edges of a simple graph with no component isomorphic to K₁ or K₂, in such a way that the graph becomes irregular, i.e., the weight sums at the vertices become pairwise distinct. The minimum of the largest weights assigned over all such irregular assignments on the vertex-disjoint union of complete graphs is determined. The method of proof also yields the smallest possible total increase in the sum of edge weights in irregular asignments, called irregularity cost.     

Improved lower bounds on k-independence

A vertex set Y in a (hyper)graph is called k-independent if in the sub(hyper)-graph induced by Y every vertex is incident to less than k edges. We prove a lower bound for the maximum cardinality of a k-independent set—in terms of degree sequences—which strengthens and generalizes several previously known results, including Turán's theorem.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$      

Upper bound for transversals of tripartite hypergraphs

We prove 2 7/9v for 3-partite hypergraphs. (This is an improvement of the trivial bound 3v.)     

Nearly uniform distribution of edges among k-subgraphs of a graph

We investigate the behavior of the function f = f(n, k, e) defined as the smallest integer with the following property: If in a graph on n vertices, the numbers of edges in any two induced subgraphs on k vertices differ by at most e, then the graph or its complement has at most f edges. One of the results states that . © 1929 John Wiley & Sons, Inc.     

Precoloring extension. I. Interval graphs

This paper is the first article in a series devoted to the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs with a bounded treewidth.     

Algorithmic complexity of list colorings

Given a graph G = (V,E) and a finite set L(v) at each vertex v ε V, the List Coloring problem asks whether there exists a function f:V → _vεVL(V) such that (i) f(v)εL(v) for each vεV and (ii) f(u) ≠f(v) whenever u, vεV and uvεE. One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L(v) has at most three elements, (2) each “color” xε_vεVL(v) occurs in at most three sets L(v), (3) each vertex vεV has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.