Rémy Belmonte, Archontia Giannopoulou, Daniel Lokshtanov, Dimitrios M. Thilikos
We study the structure of graphs that do not contain the wheel on 5 vertices W4 as an immersion, and show that these graphs can be constructed via 1, 2, and 3-edge-sums from subcubic graphs and graphs of bounded treewidth.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $γ$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an immersion and is embedded in a surface of Euler genus $γ$ has either "small" treewidth (bounded by a function of $H$ and $γ$) or "small" edge connectivity (bounded by the maximum degree of $H$). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation.
We consider the problem of how much edge connectivity is necessary to force a graph G to contain a fixed graph H as an immersion. We show that if the maximum degree in H is D, then all the examples of D-edge connected graphs which do not contain H as a weak immersion must have a tree-like decomposition called a tree-cut decomposition of bounded width. If we consider strong immersions, then it is easy to see that there are arbitrarily highly edge connected graphs which do not contain a fixed clique K_t as a strong immersion. We give a structure theorem which roughly characterizes those highly e…
We present a construction of two infinite graphs $G_1$ and $G_2$, and of an infinite set $\mathscr{F}$ of graphs such that $\mathscr{F}$ is an antichain with respect to the immersion relation and, for each graph $G$ in $\mathscr{F}$, both $G_1$ and $G_2$ are subgraphs of $G$, but no graph properly immersed in $G$ admits an immersion of $G_1$ and of $G_2$. This shows that the class of infinite graphs ordered by the immersion relation does not have the finite intertwine property.
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall.
Totally odd immersions in line graphs
本文证明了常重数多重图的线图满足完全奇浸入版本的Hadwiger猜想，推广了Guyer和McDonald的结果。
DeVos and Malekian gave a structural description of graphs avoiding an immersion of K_{3,3}, showing that all such graphs are composed over small edge-cuts from graphs with at most 8 vertices and from 3-regular planar graphs. We provide another proof of this fact, simpler in some aspects.
This paper gives a precise structure theorem for the class of graphs which do not contain $W_4$ as an immersion. This strengthens a previous result of Belmonte at al. that gives a rough description of this class. In fact, we prove a stronger theorem concerning rooted immersions of $W_4$ where one terminal is specified in advance. This stronger result is key in a forthcoming structure theorem for graphs with no $K_{3,3}$ immersion.
Splitter Theorems for Graph Immersions
本文针对k边连通图（k为偶数）及3边连通且内部4边连通图建立图浸入的拆分定理，并由此证明任意至少七顶点且可浸入K_5的该类图必可浸入K_{3,3}。
The immersion number of a graph $G$, denoted im$(G)$, is the largest $t$ such that $G$ has a $K_t$-immersion. In this note we are interested in determining the immersion number of the $m$-Mycielskian of $G$, denoted $μ_m(G)$. Given the immersion number of $G$ we provide a lower bound for im$(μ_m(G))$. To do this we introduce the "distinct neighbor property" of immersions. We also include examples of classes of graphs where im$(μ_m(G))$ exceeds the lower bound. We conclude with a conjecture about im$(μ_m(K_t))$.
A simple proof of the existence of complete bipartite graph immersion in graphs with independence number two
针对独立数为2的图，本文用更简单的论证重新证明了其包含指定顶点数的所有完全二部图浸入。
In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with $t$ vertices can be immersed in any $t$-chromatic graph, is studied. First, we present a general result about immersions and prove that the conjecture holds for graphs whose complement does not contain any induced cycle of length four and also for graphs having the property that every set of five vertices induces a subgraph with at least six edges. Then, we study the class of all graphs with …
The immersion-minimal infinitely edge-connected graph
本文刻画无限边连通图的极小障碍，证明其在浸入子式意义下有唯一极小图 halved Farey graph，而拓扑子式意义下的极小代表列表不可数。
A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected planar subcubic graph on $h>0$ edges, $G$ is a graph, and $k$ is a non-negative integer, then either $G$ contains $k$ vertex/edge-disjoint subgraphs, each containing $H$ as an immersion, or $G$ contains a set $F$ of $f(k,h)$ vertices/edges such that $G\setminus F$ does not contain $H$ as an immersion.
An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.
A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $ϕ: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $ϕ(u)$ and $ϕ(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel, and independently by Abu-Khzam and Langston, states that every graph $G$ contains $K_{χ(G)}$ as an immersion. We prove that for any constant $\varepsilon>0$ and integers $s,t\ge2$, there exists $d_0=d_0(\varepsilon,s,t)$ such that…
Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$.
We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.
A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph $H$, there exists a function $f$ such that for every 4-edge-connected graph $G$, either $G$ contains $k$ pairwise edge-disjoint subgraphs each containing $H$ as an immersion, or there exists a set of at most $f(k)$ edges of $G$ intersecting all such subgraphs. …
A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$ contains $H$ as an immersion, then for every integer $k$, the number of vertices of degree at least $k$ in $G$ is at least the number of vertices of degree at least $k$ in $H$. In this paper, we prove that this obvious necessary condition is "nearly" sufficient for graphs with no edge-cut of order 3: for every graph $H$, every $H$-immersion free graph with …
Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t + 4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no stable set of size three contains $K_{2\lfloor n/5 \rfloor}$ as an immersion.
The structure of group-labeled graphs forbidding an immersion
该研究证明了禁止特定浸入的群标记图具有特定的树割分解结构。
Totally odd immersions of complete graphs in graph products
研究完全图在图积中的完全奇浸入问题，并证明无法从四种标准图积中获得奇Hadwiger猜想的浸入版本的最小反例。
The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner obtained a precise description of all graphs with no $K_{3,3}$ minor and all graphs with no $K_5$ minor. Similar results have been achieved for the class of graphs with no $H$-minor for a number of small graphs $H$. In this paper we give a precise structure theorem for graphs which do not contain $K_{3,3}$ as an immersion. This strengthens an earlier theorem of Giannopoulou, Kamiński, and Thilikos that gives a rough description of the class of gra…
Let $G,H$ be graphs and $G*H$ represent a particular graph product of $G$ and $H$. We define $im(G)$ to be the largest $t$ such that $G$ has a $K_t$-immersion and ask: given $im(G)=t$ and $im(H)=r$, how large is $im(G*H)$? Best possible lower bounds are provided when $*$ is the Cartesian or lexicographic product, and a conjecture is offered for each of the direct and strong products, along with some partial results.
Every properly colored graph with $χ(G)=k$ colors has edge-disjoint Kempe "backbones", Kempe chains anchored by color-critical vertices for each pair of colors. Certain color permutations arrange these backbones into a clique-like structure, a strengthening of strong immersions of complete graphs. This strengthened immersion is suggested as a template for identifying the disjoint subgraphs comprising Hadwiger's conjectured $K_k$ minor present in $k$-chromatic graphs.
We prove that if $L(G)$ immerses $K_t$ then $L(mG)$ immerses $K_{mt}$, where $mG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel edge of multiplicity $m$. This implies that when $G$ is a simple graph, $L(mG)$ satisfies a conjecture of Abu-Khzam and Langston. We also show that when $G$ is a line graph, $G$ has a $K_t$-immersion iff $G$ has a $K_t$-minor whenever $t\leq 4$, but this equivalence fails in both directions when $t \geq 5$.
In 1989, Lescure and Meyniel proved, for $d=5, 6$, that every $d$-chromatic graph contains an immersion of $K_d$, and in 2003 Abu-Khzam and Langston conjectured that this holds for all $d$. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for $d = 7$. In each proof, the $d$-chromatic assumption was not fully utilized, as the proofs only use the fact that a $d$-critical graph has minimum degree at least $d - 1$. DeVos, Dvořák, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree $d-1$ has an immersion of $K_d$ fails for $d=10$ a…
An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and $f(v)$. The immersion is strong if the paths $P_{uv}$ are internally disjoint from $f(V(H))$. It is proved that for every positive integer $t$, every simple graph of minimum degree at least $200t$ contains a strong immersion of the complete graph $K_t$. For dense graphs one can say even more. If the graph has order $n$ and has $2cn^2$ edges, then there is a…
Well-quasi-ordering digraphs with no long alternating paths by the strong immersion relation
本文针对有向图在强浸入关系下的良拟序问题，证明不含方向改变k次的路径的有向图类（含顶点良拟序标记情形）是良拟序的，且该结果对子图封闭类最优。
Isoradial embeddings of planar graphs play a crucial role in the study of several models of statistical mechanics, such as the Ising and dimer models. Kenyon and Schlenker give a combinatorial characterization of planar graphs admitting an isoradial embedding, and describe the space of such embeddings. In this paper we prove two results of the same type for generalizations of isoradial embeddings: isoradial immersions and minimal immersions. We show that a planar graph admits a flat isoradial immersion if and only if its train-tracks do not form closed loops, and that a bipartite graph has a m…
How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $2^{t+\log^2 t}n$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$.…
Odd minors or odd immersions in graphs with independence number two
针对独立数为2的图，在Kühn等人反驳奇Hadwiger猜想的背景下，证明这类图必包含K_{⌈n/2⌉}作为奇子式或全奇浸入，完善了相关图论结论。
Unavoidable immersions of 4- and $f(t)$-edge-connected graphs
该研究提出环分解工具，证明足够大的4边连通图含双环浸入、线性边连通大图含$C_{t,r}$浸入，并以此推导高连通线图的不可避免子式定理。
Immersions of directed graphs in tournaments
本文研究锦标赛中有向图的浸入问题，证明了任意具有O(k)个顶点的锦标赛必含k个顶点的传递锦标赛的1-浸入，并进一步证明最小出度至少为Ck的锦标赛必含k个顶点的完全有向图的2-浸入，且这些界在相应意义下都是紧的。
Hadwiger and Hajós conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for $H$-immersion free gr…
Fix g>1. Every graph of large enough tree-width contains a g x g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g x g grid as an immersion (and hence contains any fixed graph with maximum degree at most four as an immersion). This result has a number of applications.
A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge-disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion.
The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting…
Clique immersions and independence number
本文研究图的团浸入与独立数的关系，针对所有独立数不小于3的情形，改进了 Gauthier、Le 和 Wollan 关于团浸入规模的已有下界，给出了更优的顶点数保证。
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every edge $e$ of $G$. We construct two infinite families of minimal non-1-planar graphs and show that for every integer $n > 62$, there are at least $2^{(n-54)/4}$ nonisomorphic minimal non-1-planar graphs of order $n$. It is also proved that testing 1-planarity is NP-complete.
Immersions and Albertson's conjecture
本文证明了Lescure-Meyniel关于弱浸入的猜想在顶点数至多$(1.64-o(1))k$的图上成立，并由此推进了Albertson猜想，即色数为k的图其交叉数至少为$cr(K_k)$。
Coloring Graphs With No Totally Odd Clique Immersion
证明无完全奇K_t浸入的图是O(t)-可着色的，并给出固定参数可解算法找到分解。
We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a crossing, these embeddings can also be thought of as transversal immersions of cubic graphs which we refer to as the {\it extracted graphs}. We also consider quadrangulations of the disk in which interior vertices have degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all such quadrangulations of the disk. Then, we provide four methods for …
Clique immersions in graphs of independence number two with certain forbidden subgraphs
本文研究独立数为2的图中的Lescure-Meyniel猜想，证明无特定长度洞的图及无某些小禁用子图的图均满足该浸入猜想。
Biclique immersions in graphs with independence number 2
本研究针对独立数为2的图，证明了每个此类n顶点图都包含所有具有⌈n/2⌉个顶点的完全二部图作为浸入子图，推进了浸入关系下Hadwiger猜想相关问题的研究。
A grid theorem for strong immersions of walls
本研究提出图的强浸入墙的网格定理，证明图含大墙强浸入子式等价于其无小宽度的树割分解，宽度由粘合度和躯干路径相似性度量。
In the final paper of the Graph Minors series N. Robertson and P. Seymour proved that graphs are well-quasi-ordered under the immersion ordering. A direct implication of this theorem is that each class of graphs that is closed under taking immersions can be fully characterized by forbidding a finite set of graphs (immersion obstruction set). However, as the proof of the well-quasi-ordering theorem is non-constructive, there is no generic procedure for computing such a set. Moreover, it remains an open issue to identify for which immersion-closed graph classes the computation of those sets can …
Homomorphism counting for immersion-closed classes is not isomorphism
本文证明了对于任何proper immersion闭的图类M，存在非同构的图G和H，其同构计数在M上无法区分。
Counterexamples to Clique Immersion Conjecture for Direct Products
针对Collins等人提出的关于图直积的团浸入猜想，构造了一个无限连通二分图族作为反例。
Finding hypergraph immersion is fixed-parameter tractable
本文将图浸入子式概念扩展到超图，证明超图浸入判定是固定参数可解的，给出 $O(N^6)$ 算法，并提出对偶超图浸入问题及其复杂度特征。
A graph H is strongly immersed in G if G is obtained from H by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge-disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We show that there exists a function d:N->N such that for all graphs H and G, if G contains a strong immersion of the star K_{1,d(Delta(H))|V(H)|…
Immersions of large cliques in graphs with independence number 2 and bounded maximum degree
本文研究独立数为2的图中大团浸入问题，证明在最大度及团覆盖数的相应界值下，图必含 $K_{\chi(G)}$ 和 $K_{\lceil n/2\rceil}$ 的浸入，从而验证有界最大度情形下的 Vergara 猜想。
Given graphs G and H with V(G) containing V(H), suppose that we have a u,v-path P_{uv} in G for each edge uv in H. There are obvious additional conditions that ensure that G contains H as a rooted subgraph, subdivision, or immersion; we seek conditions that ensure that G contains H as a rooted minor or minor. This naturally leads to studying sets of paths that form an H-immersion, with the additional property that paths that contain the same vertex must have a common endpoint. We say that $H$ is contractible if, whenever G contains such an H-immersion, G must also contain a rooted H-minor. We …
Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width
本文证明了具有受限刻痕宽度的欧拉图类在强浸入下是良quis序的，并提出了一种新的强浸入概念。同时指出无界度数的欧拉图类在强浸入下不满足良quis序，但弱浸入下可能满足。
Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion
本文证明可嵌入任意曲面的最大度至多 4 的欧拉有向图在强浸入下良拟序，并给出圆图、算法与 Erdős-Pósa 性质等应用。
Topological cliques in sparse expanders
本文研究稀疏扩展图中团浸入与团细分的存在性，给出三类充分条件下大团浸入或细分的渐近构造，并部分回答了 Dvořák 和 Yepremyan 关于最小度与团浸入的问题。
The Abu-Khzam$\unicode{x2013}$Langston Conjecture for Graphs with $α(G) = 2$
本文证明了 Abu-Khzam–Langston 猜想对于独立数为 2 的图类成立，并引入了图的环匹配着色概念，证明了在此设定下 Vizing 定理的一个加强形式。
Totally odd subdivisions in Kneser graphs
本文加强了 Kneser 图中关于 Odd Hadwiger Conjecture 的已有结果，证明了存在包含任意大完全全奇分拆的 $t$-色 Kneser 图，且所有 Kneser 图均含 $K_{χ(G)}$ 的全奇分拆，并扩展至全奇浸入研究。
Near-Optimal Algorithms for Maximal Clique Enumeration in Structurally Sparse Graphs
针对排除团子式和排除团浸入的图，提出两种最大团枚举算法，分别将时间复杂度优化为n·4^(2t/5+o(t))和n·3^(t/3+o(t))，并证明其指数基为渐近最优。
We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum degree at least t-1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high in- and outdegree which do not contain an immersion of the complete digraph on 3 vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construc…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。