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Register a new userThe 6-girth-thickness of the complete graph
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Christian Rubio-Montiel PhD
Publication date
2017
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English
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The $g$-girth-thickness $theta(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we determine the $6$-girth-thickness $theta(6,K_n)$ of the complete graph $K_n$ in almost all cases. And also, we calculate by computer the missing value of $theta(4,K_n)$.
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Let $ Pi_q $ be the projective plane of order $ q $, let $psi(m):=psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ ain { 3,4,dots,tfrac{q}{2}+1 } $ and $ m_a=(q+1)^2-a $. In this paper, we improve the upper bound of $ psi(m) $ given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89--97] and Jamison [Discrete Math. 74 (1989), 99--115] in the following values: if $ xgeq 2 $ is an integer and $min {4x^2-x,dots,4x^2+3x-3}$ then $psi(m) leq 2x(m-x-1)$. On the other hand, if $ q $ is even and there exists $ Pi_q $ we give a complete edge-colouring of $ K_{m_a} $ with $(m_a-a)q$ colours. Moreover, using this colouring we extend the previous results for $a={-1,0,1,2}$ given by Araujo-Pardo et al. in [J Graph Theory 66 (2011), 89--97] and [Bol. Soc. Mat. Mex. (2014) 20:17--28] proving that $psi(m_a)=(m_a-a)q$ for $ ain {3,4,dots,leftlceil frac{1+sqrt{4q+9}}{2}rightrceil -1 } $.
We show that the abelian girth of a graph is at least three times its girth. We prove an analogue of the Moore bound for the abelian girth of regular graphs, where the degree of the graph is fixed and the number of vertices is large. We conclude that one could try to improve the Moore bound for graphs of fixed degree and many vertices by trying to improve its analogue concerning the abelian girth.
Let $H_{mathrm{WR}}$ be the path on $3$ vertices with a loop at each vertex. D. Galvin conjectured, and E. Cohen, W. Perkins and P. Tetali proved that for any $d$-regular simple graph $G$ on $n$ vertices we have $$hom(G,H_{mathrm{WR}})leq hom(K_{d+1},H_{mathrm{WR}})^{n/(d+1)}.$$ In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali. Our main tool is a simple bijection between the Widom-Rowlinson model and the hard-core model on another graph. We also give a large class of graphs $H$ for which we have $$hom(G,H)leq hom(K_{d+1},H)^{n/(d+1)}.$$ In particular, we show that the above inequality holds if $H$ is a path or a cycle of even length at least $6$ with loops at every vertex.
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 $t$.
A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s geq 3$ and $t geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $Theta ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s geq 4$ and $t geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $Omega( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer a question of Chakraborti and Loh. General estimates on the number of copies of $K_{a,b}$ in a $K_s$-saturated graph are also obtained, but finding an asymptotic formula remains open.
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