arXiv subjects

Gexin Yu

Publications and source records attributed to Gexin Yu.

At least 37 records · Page 2Linked to original sources

arXiv2019

Planar graphs with girth at least 5 are (3,4)-colorable

A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ nonempty subsets so that the subgraph induced by the $i$th part has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It is known that for each pair $(d_1, d_2)$, there exists a planar graph with girth $4$ that is not $(d_1, d_2)$-colorable. This sparked the interest in finding the pairs $(d_1, d_2)$ such that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable. Given $d_1\leq d_2$, it is known that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable if either $d_1\geq 2$ and $d_1+d_2\geq 8$ or $d_1=1$ and $d_2\geq 10$. We improve an aforementioned result by providing the first pair $(d_1, d_2)$ in the literature satisfying $d_1+d_2\leq 7$ where planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable. Namely, we prove that planar graphs with girth at least $5$ are $(3, 4)$-colorable.

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arXiv2019

Fat-triangle linkage and kite-linked graphs

For a multigraph $H$, a graph $G$ is $H$-linked if every injective mapping $ϕ: V(H)\to V(G)$ can be extended to an $H$-subdivision in $G$. We study the minimum connectivity required for a graph to be $H$-linked. A $k$-fat-triangle $F_k$ is a multigraph with three vertices and a total of $k$ edges. We determine a sharp connectivity requirement for a graph to be $F_k$-linked. In particular, any $k$-connected graph is $F_k$-linked when $F_k$ is connected. A kite is the graph obtained from $K_4$ by removing two edges at a vertex. As a nontrivial application of $F_k$-linkage, we then prove that every $8$-connected graph is kite-linked, which shows that the required connectivity for a graph to be kite-linked is $7$ or $8$.

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arXiv2019

Minimum degree condition for a graph to be knitted

For a positive integer $k$, a graph is $k$-knitted if for each $k$-subset $S$ of vertices, and every partition of $S$ into disjoint parts $S_1, \ldots, S_t$ for some $t\ge 1$, one can find disjoint connected subgraphs $C_1, \ldots, C_t$ such that $C_i$ contains $S_i$ for each $i$. In this article, we show that if the minimum degree of an $n$-vertex graph $G$ is at least $n/2+k/2-1$ when $n\ge 2k+3$, then $G$ is $k$-knitted. The minimum degree is sharp. As a corollary, we obtain that $k$-contraction-critical graphs are $\left\lceil\frac{k}{8}\right\rceil$-connected.

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arXiv2018

DP-4-colorability of two classes of planar graphs

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is DP-$4$-colorable for $k=5$ and $6$. As a consequence, we obtain two new classes of $4$-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.

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arXiv2018

DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8\}$ is DP-$3$-colorable, extending three existing results on $3$-choosability of planar graphs.

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arXiv2018

DP-3-coloring of some planar graphs

In this article, we use a unified approach to prove several classes of planar graphs are DP-$3$-colorable, which extend the corresponding results on $3$-choosability.

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arXiv2018

Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by $χ'_{s}(G)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erdős and Nešetřil conjectured that $χ'_{s}(G)\leq\frac{5}{4}Δ(G)^2$, where $Δ(G)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.

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arXiv2018

Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^Δ\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge d_1$ are $3$-choosable, where $d^Δ$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{4,5\}$-cycles and $d^Δ\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge 2$ are DP-$3$-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, $d_0\le 3$ and $d_1\le 2$. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.

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arXiv2018

Planar graphs without 4-cycles and close triangles are (2,0,0)-colorable

For a set of nonnegative integers $c_1, \ldots, c_k$, a $(c_1, c_2,\ldots, c_k)$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, \ldots, V_k$ such that for every $i$, $1\le i\le k, G[V_i]$ has maximum degree at most $c_i$. We prove that all planar graphs without 4-cycles and no less than two edges between triangles are $(2,0,0)$-colorable.

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arXiv2018

Strong chromatic index of graphs with maximum degree four

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $Δ$ has a strong edge-coloring using at most $\frac{5}{4}Δ^2$ colors if $Δ$ is even, and at most $\frac{5}{4}Δ^2 - \frac{1}{2}Δ+ \frac{1}{4}$ if $Δ$ is odd. Despite recent progress for large $Δ$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $Δ= 3$, leaving the need for new approaches to verify the conjecture for any $Δ\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.

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arXiv2018

Covering 2-connected 3-regular graphs with disjoint paths

A path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number $p(G)$ of graph $G$ is the cardinality of a path cover with the minimum number of paths. Reed in 1996 conjectured that a $2$-connected $3$-regular graph has path cover number at most $\lceil n/10\rceil$. In this paper, we confirm this conjecture.

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arXiv2018

The strong chromatic index of $(3,Δ)$-bipartite graphs

A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. We study bipartite graphs with one part having maximum degree at most $3$ and the other part having maximum degree $Δ$. We show that every such graph has a strong edge-coloring using at most $3 Δ$ colors. Our result confirms a conjecture of Brualdi and Quinn Massey ~\cite{[BQ]} for this class of bipartite graphs.

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arXiv2018

Maximum average degree and relaxed coloring

We say a graph is $(d, d, \ldots, d, 0, \ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\ldots,O_b$ and $a$ sets $D_1, D_2,\ldots, D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad(G)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a, b$, we show that if $mad(G)< \frac{4}{3}a + b$, then $G$ is $(1_1, 1_2, \ldots, 1_a, 0_1, \ldots, 0_b)$-colorable.

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arXiv2016

Equitable coloring of sparse planar graphs

A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $χ_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\ge m$. We show that for planar graphs $G$ with minimum degree at least two, $χ_{eq}^*(G)\le 4$ if the girth of $G$ is at least $10$, and $χ_{eq}^*(G)\le 3$ if the girth of $G$ is at least $14$.

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arXiv2016

Extremal permutations in routing cycles

Let $G$ be a graph on $n$ vertices, labeled $v_1,\ldots,v_n$ and $π$ be a permutation on $[n]:=\{1,2,\cdots, n\}$. Suppose that each pebble $p_i$ is placed at vertex $v_{π(i)}$ and has destination $v_i$. During each step, a disjoint set of edges is selected and the pebbles on each edge are swapped. Let $rt(G, π)$, the routing number for $π$, be the minimum number of steps necessary for the pebbles to reach their destinations. Li, Lu, and Yang prove that $rt(C_n, π)\le n-1$ for any permutation on $n$-cycle $C_n$ and conjecture that for $n \geq 5$, if $rt(C_n, π) = n-1$, then $π= (123\cdots n)$ or its inverse. By a computer search, they show that the conjecture holds for $n<8$. We prove in this paper that the conjecture holds for all even $n$.

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arXiv2015

A relaxation of the strong Bordeaux Conjecture

Let $c_1, c_2, \cdots, c_k$ be $k$ non-negative integers. A graph $G$ is $(c_1, c_2, \cdots, c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, \ldots, V_k$, such that the subgraph $G[V_i]$, induced by $V_i$, has maximum degree at most $c_i$ for $i=1, 2, \ldots, k$. Let $\mathcal{F}$ denote the family of plane graphs with neither adjacent 3-cycles nor $5$-cycle. Borodin and Raspaud (2003) conjectured that each graph in $\mathcal{F}$ is $(0,0,0)$-colorable. In this paper, we prove that each graph in $\mathcal{F}$ is $(1, 1, 0)$-colorable, which improves the results by Xu (2009) and Liu-Li-Yu (2014+).

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arXiv2015

A relaxation of the Bordeaux Conjecture

A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $φ:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph without intersecting triangles and $5$-cycles is $3$-colorable. We prove in this paper that every planar graph without intersecting triangles and $5$-cycles is (2,0,0)-colorable.

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arXiv2014

Planar graphs without 5-cycles and intersecting triangles are $(1,1,0)$-colorable

A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $φ:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph without $5$-cycles and intersecting triangles is $(0,0,0)$-colorable. We prove in this paper that such graphs are $(1,1,0)$-colorable.

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