Every Graph $G$ is Hall $\Delta(G)$-Extendible

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.

Completing Partial Proper Colorings using Hall's Condition

In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph $G$ with list assignment $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $m$-completable for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable.

Relaxations of Hall’s Condition: Optimal batch codes with multiple queries

Combinatorial batch codes model the storage of a database on a given number of servers such that any k or fewer items can be retrieved by reading at most t items from each server. A combinatorial batch code with parameters n; k; m; t can be represented by a system F of n (not necessarily distinct) sets over an m-element underlying set X, such that for any k or fewer members of F there exists a system of representatives in which each element of X occurs with multiplicity at most t. The main purpose is to determine the minimum N(n; k; m; t) of total data storage ?F?F |F| over all combinatorial batch codes F with given parameters. Previous papers concentrated on the case t = 1. Here we obtain the first nontrivial results on combinatorial batch codes with t > 1. We determine N(n; k; m; t) for all cases with k ? 3t, and also for all cases where n ? t? m dk=te?2?. Our results can be considered equivalently as minimum total size ?F?F |F| over all set systems F of given order m and size n, which satisfy a relaxed version of Hall's Condition; that is, |UF?| ? |F?|/t holds for every subsystem F? ? F of size at most k.

HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.