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Register a new userMinimum k-critical bipartite graphs
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Added by
Karol Suchan
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, Delta_U, Delta_V)$, where $Delta_U$ and $Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates $G$ that has a complete matching, i.e., a matching of size $m$. We show that, if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with $a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an $(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a simple construction that creates such a $k$-critical bipartite graph whenever possible. Our techniques are based on Halls marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
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A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason [Combinatorica 16 (1996), 123-141]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.
Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworths Theorem -- that may be of independent interest.
Recently, Kostochka and Yancey proved that a conjecture of Ore is asymptotically true by showing that every $k$-critical graph satisfies $|E(G)|geqleftlceilleft(frac{k}{2}-frac{1}{k-1}right)|V(G)|-frac{k(k-3)}{2(k-1)}rightrceil.$ They also characterized the class of graphs that attain this bound and showed that it is equivalent to the set of $k$-Ore graphs. We show that for any $kgeq33$ there exists an $varepsilon>0$ so that if $G$ is a $k$-critical graph, then $|E(G)|geqleft(frac{k}{2}-frac{1}{k-1}+varepsilon_kright)|V(G)|-frac{k(k-3)}{2(k-1)}-(k-1)varepsilon T(G)$, where $T(G)$ is a measure of the number of disjoint $K_{k-1}$ and $K_{k-2}$ subgraphs in $G$. This also proves for $kgeq33$ the following conjecture of Postle regarding the asymptotic density: For every $kgeq4$ there exists an $varepsilon_k>0$ such that if $G$ is a $k$-critical $K_{k-2}$-free graph, then $|E(G)|geq left(frac{k}{2}-frac{1}{k-1}+varepsilon_kright)|V(G)|-frac{k(k-3)}{2(k-1)}$. As a corollary, our result shows that the number of disjoint $K_{k-2}$ subgraphs in a $k$-Ore graph scales linearly with the number of vertices and, further, that the same is true for graphs whose number of edges is close to Kostochka and Yanceys bound.
A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $kge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs for $k in {4,5}$ using a computer generation algorithm.
Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.
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