Graph coloring satisfying restraints

AbstractFor an integer k⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v)≠r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k≠3, there are infinitely many amenable k-critical graphs. For k ⩾ 3, we use a construction of B. Toft and amenable graphs to associate a k-colorable graph to any k-colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property—being strongly critical—that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Hajós construction applied to a single edge in each of two strongly k-critical graphs yields an amenable graph. However, for any k⩾5, there are amenable k-critical graphs for which the Hajós construction on two copies is not amenable

On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.Comment: 17 pages, 5 figure

Spectrum of Sizes for Perfect Deletion-Correcting Codes

One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the Levenshte\u{\i}n distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect $t$-deletion-correcting code, given the length $n$ and the alphabet size~$q$. In this paper, we determine completely the spectrum of possible sizes for perfect $q$-ary 1-deletion-correcting codes of length three for all $q$, and perfect $q$-ary 2-deletion-correcting codes of length four for almost all $q$, leaving only a small finite number of cases in doubt.Comment: 23 page

On a generalized matching problem arising in estimating the eigenvalue variation of two matrices

Elsner L, Johnson CR, Ross JA, Schönheim J. On a generalized matching problem arising in estimating the eigenvalue variation of two matrices. European Journal of Combinatorics. 1983;4:133-136