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A graph $G$ contains $H$ as an emph{immersion} if there is an injective mapping $phi: V(H)rightarrow V(G)$ such that for each edge $uvin E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $phi(u)$ and $phi(v)$, and all the paths $P_{uv}$, $uvin E(H)$, are pairwise edge-disjoint. An analogue of Hadwigers conjecture for the clique immersions by Lescure and Meyniel states that every graph $G$ contains $K_{chi(G)}$ as an immersion. We consider the average degree condition and prove that for any bipartite graph $H$, every $H$-free graph $G$ with average degree $d$ contains a clique immersion of order $(1-o(1))d$, implying that Lescure and Meyniels conjecture holds asymptotically for graphs without fixed bipartite graph.
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The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)geq (1+o(1))ctsqrt{ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $csim 1.276dotsc$. We improve this to $h(G) geq (1+o(1))tsqrt{log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
A recent result of Condon, Kim, K{u}hn and Osthus implies that for any $rgeq (frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost $alpha n$-regular graph $G$ with any $alpha>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into $n$-vertex trees. Moreover, this implies that for any $alpha>0$ and an $n$-vertex almost $alpha n$-regular graph $G$, with high probability, the randomly perturbed graph $Gcup mathbf{G}(n,O(frac{1}{n}))$ has an approximate decomposition into all collections of bounded degree trees of size at most $(1-o(1))n$ simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey-Turan theory and the randomly perturbed graph model.
We prove that for any $tge 3$ there exist constants $c>0$ and $n_0$ such that any $d$-regular $n$-vertex graph $G$ with $tmid ngeq n_0$ and second largest eigenvalue in absolute value $lambda$ satisfying $lambdale c d^{t}/n^{t-1}$ contains a $K_t$-factor, that is, vertex-disjoint copies of $K_t$ covering every vertex of $G$.
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.
Let $G$ be a graph whose edges are coloured with $k$ colours, and $mathcal H=(H_1,dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1le ile k$. Let $phi_{k}(n,mathcal H)$ be the smallest number $phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $mathcal H$-decomposition with at most $phi$ elements. Extending the previous results of Liu and Sousa [Monochromatic $K_r$-decompositions of graphs, Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $mathcal H$ is a clique and $nge n_0(mathcal H)$ is sufficiently large.
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