arXiv subjects

Zi-Xia Song

Publications and source records attributed to Zi-Xia Song.

48 records · Page 3Linked to original sources

arXiv2018

Multicolor Gallai-Ramsey numbers of $C_9$ and $C_{11}$

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this paper, we study Gallai-Ramsey numbers of odd cycles. We prove that for $n\in\{4,5\}$ and all $k\ge1$, $GR_k(C_{2n+1})= n\cdot 2^k+1$. This new result provides partial evidence for the first two open cases of the Triple Odd Cycle Conjecture of Bondy and Erdős from 1973. Our technique relies heavily on the structural result of Gallai on Gallai colorings of complete graphs. We believe the method we developed can be used to determine the exact values of $GR_k(C_{2n+1})$ for all $n\ge6$.

math.CO↗

arXiv2018

Planar anti-Ramsey numbers of matchings

Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. In this paper we study planar anti-Ramsey numbers of matchings. For all $t\ge1$, let $M_t$ denote a matching of size $t$. We prove that for all $t\ge6$ and $n\ge 3t-6$, $2n+3t-15\le ar_{_{\mathcal{P}}}(n, {M}_t)\le 2n+4t-14$, which significantly improves the existing lower and upper bounds for $ar_{_\mathcal{P}}(n, M_t)$. It seems that for each $t\ge6$, the lower bound we obtained is the exact value of $ar_{_{\mathcal{P}}}(n, {M}_t)$ for sufficiently large $n$. This is indeed the case for $M_6$. We prove that $ar_{_\mathcal{P}}(n, M_6)=2n+3$ for all $n\ge30$.

math.CO↗

arXiv2017

Planar anti-Ramsey numbers for paths and cycles

Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. Analogous to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number of $H$ is the maximum number of edges of a planar graph on $n$ vertices without containing $H$ as a subgraph. The study of $ar_{_\mathcal{P}}(n, H)$ (under the name of rainbow numbers) was initiated by Horňák, Jendrol$'$, Schiermeyer and Soták [J Graph Theory 78 (2015) 248--257]. In this paper we study planar anti-Ramsey numbers for paths and cycles. We first establish lower bounds for $ar_{_\mathcal{P}}(n, P_k)$ when $n\ge k\ge8$. We then improve the existing lower bound for $ar_{_\mathcal{P}}(n, C_k)$ when $k\geq 5$ and $n\geq k^2-k$. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for $ar_{_\mathcal{P}}(n, C_6)$ when $n\ge8$ and $ar_{_\mathcal{P}}(n, C_7)$ when $n\geq 13$, respectively.

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arXiv2017

Star chromatic index of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $χ'_s(G)\le3$ for an arbitrary graph $G$. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs $G$ with $δ(G)\le2$ such that $χ'_s(G)>k$ but $χ'_s(G-v)\le k$ for any $v\in V(G)$, where $k\in\{5,6\}$. We finally apply the structure results, along with a simple discharging method, to prove that every subcubic multigraph $G$ is star $6$-edge-colorable if $mad(G)<5/2$, and star $5$-edge-colorable if $mad(G)<24/11$, respectively, where $mad(G)$ is the maximum average degree of a multigraph $G$. This partially confirms the conjecture of Dvořák, Mohar and Šámal.

math.CO↗

arXiv2017

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.

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arXiv2017

Clique Minors in Double-critical Graphs

A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erdős and Lovász that the complete graphs are the only double-critical $t$-chromatic graphs remains open for all $t\ge6$. Given the difficulty in settling Erdős and Lovász's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every double-critical $t$-chromatic graph contains a $K_t$ minor and verified their conjecture for $t\le7$. Albar and Gonçalves recently proved that every double-critical $8$-chromatic graph contains a $K_8$ minor, and their proof is computer-assisted. In this paper we prove that every double-critical $t$-chromatic graph contains a $K_t$ minor for all $t\le9$. Our proof for $t\le8$ is shorter and computer-free.

math.CO↗

arXiv2017

Gallai-Ramsey numbers of $C_9$ with multiple colors

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\cdot 2^k+1$ for all $k\ge1$. This new result provides partial evidence for the first open case of the Triple Odd Cycle Conjecture of Bondy and Erdős from 1973. Our technique relies heavily on the structural result of Gallai on edge-colorings of complete graphs without rainbow triangles. We believe the method we developed can be used to determine the exact values of $gr_k(K_3, C_n)$ for odd integers $n\ge11$.

math.CO↗

arXiv2017

Antimagic orientations of even regular graphs

A labeling of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1, \ldots, m\}$. A labeling of $D$ is antimagic if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz [On antimagic directed graphs, J Graph Theory 64 (2010) 219--232] initiated the study of antimagic labelings of digraphs, and conjectured that every connected graph admits an antimagic orientation, where an orientation $D$ of a graph $G$ is antimagic if $D$ has an antimagic labeling. It remained unknown whether every disjoint union of cycles admits an antimagic orientation. In this paper, we first answer this question in the positive by proving that every $2$-regular graph has an antimagic orientation. We then show that for any integer $d\ge2$, every connected, $2d$-regular graph has an antimagic orientation. Our technique is new.

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arXiv2017

Hadwiger's conjecture for graphs with forbidden holes

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\ge χ(G)$, where $χ(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $α(G)\ge3$ has no hole of length between $4$ and $2α(G)-1$, then $h(G)\geχ(G)$. We also prove that if a graph $G$ with independence number $α(G)\ge2$ has no hole of length between $4$ and $2α(G)$, then $G$ contains an odd clique minor of size $χ(G)$, that is, such a graph $G$ satisfies the odd Hadwiger's conjecture.

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arXiv2017

Double-critical graph conjecture for claw-free graphs

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erd\H os and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph $G$ with chromatic number $t$ is double-critical, then $G$ is the complete graph on $t$ vertices. This has been verified for $t \le 5$, but remains open for $t \ge 6$. In this paper, we first prove that if $G$ is a non-complete, double-critical graph with chromatic number $t \ge 6$, then no vertex of degree $t + 1$ is adjacent to a vertex of degree $t+1$, $t + 2$, or $t + 3$ in $G$. We then use this result to show that the Double-Critical Graph Conjecture is true for double-critical graphs $G$ with chromatic number $t \le 8$ if $G$ is claw-free.

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arXiv2016

Coloring Graphs with Forbidden Minors

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gonçalves and generalize it to the next step by showing that every graph with no $K_t$ minor is $(2t-6)$-colorable, where $t\in\{7,8,9\}$. We then prove that graphs with no $K_8^-$ minor are $9$-colorable and graphs with no $K_8^=$ minor are $8$-colorable. Finally we prove that if Mader's bound for the extremal function for $K_p$ minors is true, then every graph with no $K_p$ minor is $(2t-6)$-colorable for all $p\ge5$. This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest.

math.CO↗

arXiv2015

A Note on Weighted Rooted Trees

Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are unrelated. Let $ω$ be a nonnegative weight function defined on $V(T)$ with $\sum_{v\in V(T)}ω(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\sum_{v\in V(P)}ω(v)\ge \frac13$ for some $u\in V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in A }ω(a)\ge \frac13$ and $\sum_{b\in B }ω(b)\ge \frac13$. The bound $\frac13$ is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé.

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