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Register a new userOn the structure of graphs excluding $K_{4}$, $W_{4}$, $K_{2,4}$ and one other graph as a rooted minor
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Benjamin Moore
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In this paper we give structural characterizations of graphs not containing rooted $K_{4}$, $W_{4}$, $K_{2,4}$, and a graph we call $L$.
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Let $tgeq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $muleft( Gright) $ satisfies [ muleft( Gright) leqfrac{t-1}{2}+sqrt{n+frac{t^{2}-2t-3}{4}}, ] with equality if and only if $nequiv1$ $(operatorname{mod}$ $t)$ and $G=K_{1}veeleftlfloor n/trightrfloor K_{t}$. For $t=3$ the maximum $muleft( Gright) $ is found exactly for any $n>40000$.
Write $rholeft( Gright) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}veeoverline{K}_{n-r}.$ Let $n>rgeq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $rholeft( Gright) $ for $r=2$, and found an asymptotically tight bound on $rholeft( Gright) $ for $rgeq3.$ They also conjectured that [ rholeft( Gright) >rholeft( S_{n,r-1}right) , ] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.
Given graphs $G, H_1, H_2$, we write $G rightarrow ({H}_1, H_2)$ if every ${$red, blue$}$-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G rightarrow ({H}_1, H_2)$, but $G+erightarrow ({H}_1, H_2)$ for every edge $e$ in $overline{G}$. Motivated by a conjecture of Hanson and Toft from 1987, we study the minimum number of edges over all $(K_t, K_{1,k})$-co-critical graphs on $n$ vertices. We prove that for all $tge3$ and $kge 3$, there exists a constant $ell(t, k)$ such that, for all $n ge (t-1)k+1$, if $G$ is a $(K_t, K_{1,k})$-co-critical graph on $n$ vertices, then $$ e(G)ge left(2t-4+frac{k-1}{2}right)n-ell(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $tin{3, 4,5}$ and all $kge3$ and $nge (2t-2)k+1$. It seems non-trivial to construct extremal $(K_t, K_{1,k})$-co-critical graphs for $tge6$. We also obtain the sharp bound for the size of $(K_3, K_{1,3})$-co-critical graphs on $nge13$ vertices by showing that all such graphs have at least $3n-4$ edges.
For a simple graph $G$, let $chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $Delta geq 4$, if $G$ has maximum degree at most $Delta$ and is $K_{Delta}$-free, then $chi_f(G) leq Delta-tfrac{1}{8}$ unless $G= C^2_8$ or $G = C_5boxtimes K_2$. This im proves the result in [King, Lu, and Peng, SIAM J. Discrete Math., 26(2) (2012), pp. 452-471] for $Delta geq 4$ and the result in [Katherine and King, SIAM J.Discrete Math., 27(2) (2013), pp. 1184-1208] for $Delta in {6,7,8}$.
The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-partite graph. Furedi gave a short proof that one can choose alpha=epsilon. We give a bound for the relationship of alpha and varepsilon which is asymptotically sharp as epsilon to 0.
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