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Register a new userOn the 12-representability of induced subgraphs of a grid graph
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The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al. initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al. by providing a complete characterization of $12$-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area.
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A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x eq y$, alternate in $w$ if and only if $(x,y)in E$. Halld{o}rsson et al. have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary to this result is that any 3-colorable graph is word-representable. Akrobotu et al. have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs $W_5$ and $W_7$).
This exposition contains a short and streamlined proof of the recent result of Kwan, Letzter, Sudakov and Tran that every triangle-free graph with minimum degree $d$ contains an induced bipartite subgraph with average degree $Omega(ln d/lnln d)$.
In 1975 Bollobas, ErdH os, and Szemeredi asked the following question: given positive integers $n, t, r$ with $2le tle r-1$, what is the largest minimum degree $delta(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szab{o}, and Szab{o} and Tardos in 2006. In this paper we investigate the $r>t+1$ case of the problem, which has remained dormant for over forty years. We resolve the problem exactly in the case when $r equiv -1 pmod{t}$, and up to an additive constant for many other cases, including when $r geq (3t-1)(t-1)$. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$-partite $rn$-vertex graphs of chromatic number at most $t$.
In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $lambda_n$. Using a novel construction of subcomponents we study the largest component for $lambda_n=frac{1+chi_n}{n}$, where $epsilonge chi_nge n^{-{1/3}+ delta}$, $delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $chi_n=epsilon$, $| C_n^{(1)}|sim alpha(epsilon) frac{1+chi_n}{n} 2^n$ and for $o(1)=chi_nge n^{-{1/3}+delta}$, $| C_n^{(1)}| sim 2 chi_n frac{1+chi_n}{n} 2^n$ holds. This improves the result of cite{Bollobas:91} where constant $chi_n=chi$ is considered. In particular, in case of $lambda_n=frac{1+epsilon} {n}$, our analysis implies that a.s. a unique giant component exists.
For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $kin{1, 2, ..., n - 1}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.
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